{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0001","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"478b68e9f6392625a14e36de8b79a3cdbcc84073dc5a3806b685c429d97a4295","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216001,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,3]],\"point\":[0,0]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[0, 0], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 3]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([0, 0], [[-1, 0, 0, 0, Yes], [0, -1, 0, 0, Yes], [1, 1, 3, 0, Yes]], Yes)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([0, 0], [[-1, 0, 0, 0, Yes], [0, -1, 0, 0, Yes], [1, 1, 3, 0, Yes]], Yes)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0002","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce214713bc07cac6b3e6b2d7a11609894534735bec23ac5ea1051c1c6a68c206","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216002,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,4]],\"point\":[1,1]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[1, 1], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 4]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([1, 1], [[-1, 0, 0, -1, Yes], [0, -1, 0, -1, Yes], [1, 1, 4, 2, Yes]], Yes)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([1, 1], [[-1, 0, 0, -1, Yes], [0, -1, 0, -1, Yes], [1, 1, 4, 2, Yes]], Yes)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0003","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a35b7c901e723a4aa2e37f295f9d388f32b6c27281ed7f3133db9ec4e3b932b","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216003,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,5]],\"point\":[2,2]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[2, 2], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 5]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([2, 2], [[-1, 0, 0, -2, Yes], [0, -1, 0, -2, Yes], [1, 1, 5, 4, Yes]], Yes)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([2, 2], [[-1, 0, 0, -2, Yes], [0, -1, 0, -2, Yes], [1, 1, 5, 4, Yes]], Yes)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0004","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcf30cda2bb89423f8c30fc05a8b23192758e8b78165d667e0eab803b1b8a509","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216004,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,6]],\"point\":[3,3]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[3, 3], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 6]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([3, 3], [[-1, 0, 0, -3, Yes], [0, -1, 0, -3, Yes], [1, 1, 6, 6, Yes]], Yes)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([3, 3], [[-1, 0, 0, -3, Yes], [0, -1, 0, -3, Yes], [1, 1, 6, 6, Yes]], Yes)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0005","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fbd8e0e4156a81eda20446b9ac07b3961e55ab4562ff5bc2dc30ce7b5f33ec4","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216005,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,7]],\"point\":[4,0]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[4, 0], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 7]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([4, 0], [[-1, 0, 0, -4, Yes], [0, -1, 0, 0, Yes], [1, 1, 7, 4, Yes]], Yes)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([4, 0], [[-1, 0, 0, -4, Yes], [0, -1, 0, 0, Yes], [1, 1, 7, 4, Yes]], Yes)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0006","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92040aa71f508d0719d23664f6ca2d2359bf54542f2a0ab1da3710a9af34a03b","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216006,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,8]],\"point\":[0,1]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[0, 1], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 8]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([0, 1], [[-1, 0, 0, 0, Yes], [0, -1, 0, -1, Yes], [1, 1, 8, 1, Yes]], Yes)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([0, 1], [[-1, 0, 0, 0, Yes], [0, -1, 0, -1, Yes], [1, 1, 8, 1, Yes]], Yes)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0007","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c93d5b7c870caf79d16fa285fe461aca7fc6003a9b8f59c05096b57c9146dac5","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216007,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,9]],\"point\":[1,2]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[1, 2], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 9]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([1, 2], [[-1, 0, 0, -1, Yes], [0, -1, 0, -2, Yes], [1, 1, 9, 3, Yes]], Yes)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([1, 2], [[-1, 0, 0, -1, Yes], [0, -1, 0, -2, Yes], [1, 1, 9, 3, Yes]], Yes)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0008","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"990177f7e003c404f473e209ce60dd112922fe3486fa109b06ff1822dffd1c75","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216008,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,3]],\"point\":[2,3]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[2, 3], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 3]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([2, 3], [[-1, 0, 0, -2, Yes], [0, -1, 0, -3, Yes], [1, 1, 3, 5, No]], No)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([2, 3], [[-1, 0, 0, -2, Yes], [0, -1, 0, -3, Yes], [1, 1, 3, 5, No]], No)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0009","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a4e2504f681448fe43c6b68450860ed47fe87485c3806f4c865106842c74e91","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216009,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,4]],\"point\":[3,0]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[3, 0], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 4]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([3, 0], [[-1, 0, 0, -3, Yes], [0, -1, 0, 0, Yes], [1, 1, 4, 3, Yes]], Yes)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([3, 0], [[-1, 0, 0, -3, Yes], [0, -1, 0, 0, Yes], [1, 1, 4, 3, Yes]], Yes)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0010","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac2c1451c85e56946188a82ef378f63637c50000b91e7806b22ec59a84d98fa4","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216010,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,5]],\"point\":[4,1]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[4, 1], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 5]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([4, 1], [[-1, 0, 0, -4, Yes], [0, -1, 0, -1, Yes], [1, 1, 5, 5, Yes]], Yes)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([4, 1], [[-1, 0, 0, -4, Yes], [0, -1, 0, -1, Yes], [1, 1, 5, 5, Yes]], Yes)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0011","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bb22c0701ae70df53a025cacdce1024246452c4bcb263116412278075101c09","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216011,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,6]],\"point\":[0,2]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[0, 2], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 6]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([0, 2], [[-1, 0, 0, 0, Yes], [0, -1, 0, -2, Yes], [1, 1, 6, 2, Yes]], Yes)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([0, 2], [[-1, 0, 0, 0, Yes], [0, -1, 0, -2, Yes], [1, 1, 6, 2, Yes]], Yes)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0012","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"069a9be48930acbd63b6345cef5a72e1fea1fec8391327697d661810fb7c156e","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216012,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,7]],\"point\":[1,3]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[1, 3], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 7]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([1, 3], [[-1, 0, 0, -1, Yes], [0, -1, 0, -3, Yes], [1, 1, 7, 4, Yes]], Yes)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([1, 3], [[-1, 0, 0, -1, Yes], [0, -1, 0, -3, Yes], [1, 1, 7, 4, Yes]], Yes)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0013","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"daf3ccf118801384c783be3b0d52ef437059d496b08ebdcf9290d07ec55ea3f4","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216013,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,8]],\"point\":[2,0]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[2, 0], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 8]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([2, 0], [[-1, 0, 0, -2, Yes], [0, -1, 0, 0, Yes], [1, 1, 8, 2, Yes]], Yes)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([2, 0], [[-1, 0, 0, -2, Yes], [0, -1, 0, 0, Yes], [1, 1, 8, 2, Yes]], Yes)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0014","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1dbd2780f3d78b1fb8a83af765a3aa84931f6aef8135c6f6776c82645c64f0c2","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216014,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,9]],\"point\":[3,1]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[3, 1], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 9]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([3, 1], [[-1, 0, 0, -3, Yes], [0, -1, 0, -1, Yes], [1, 1, 9, 4, Yes]], Yes)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([3, 1], [[-1, 0, 0, -3, Yes], [0, -1, 0, -1, Yes], [1, 1, 9, 4, Yes]], Yes)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0015","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d312dbe86f2dbd459b4d549d1077174c46345b36df7e4a932ccbdac8a4834c5","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216015,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,3]],\"point\":[4,2]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[4, 2], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 3]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([4, 2], [[-1, 0, 0, -4, Yes], [0, -1, 0, -2, Yes], [1, 1, 3, 6, No]], No)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([4, 2], [[-1, 0, 0, -4, Yes], [0, -1, 0, -2, Yes], [1, 1, 3, 6, No]], No)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0016","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abddd008ad17506fa1f7a84e8fd064069c7f3cff10138194557da4763d216993","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216016,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,4]],\"point\":[0,3]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[0, 3], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 4]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([0, 3], [[-1, 0, 0, 0, Yes], [0, -1, 0, -3, Yes], [1, 1, 4, 3, Yes]], Yes)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([0, 3], [[-1, 0, 0, 0, Yes], [0, -1, 0, -3, Yes], [1, 1, 4, 3, Yes]], Yes)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0017","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9ebde9e0138fcbb516c92cbd42515d3f94b39f8ebf7a4d17775da7d187540ed","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216017,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,5]],\"point\":[1,0]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[1, 0], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 5]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([1, 0], [[-1, 0, 0, -1, Yes], [0, -1, 0, 0, Yes], [1, 1, 5, 1, Yes]], Yes)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([1, 0], [[-1, 0, 0, -1, Yes], [0, -1, 0, 0, Yes], [1, 1, 5, 1, Yes]], Yes)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0018","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bb127934d06b7ca8d59c458de4a44b4fa756f3d445d5401717ce70632bde16b","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216018,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,6]],\"point\":[2,1]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[2, 1], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 6]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([2, 1], [[-1, 0, 0, -2, Yes], [0, -1, 0, -1, Yes], [1, 1, 6, 3, Yes]], Yes)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([2, 1], [[-1, 0, 0, -2, Yes], [0, -1, 0, -1, Yes], [1, 1, 6, 3, Yes]], Yes)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0019","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54d95b1d270db9aee0c91c3388e0dff5fa641bf03b73a9bad5be2b41acee8c94","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216019,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,7]],\"point\":[3,2]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[3, 2], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 7]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([3, 2], [[-1, 0, 0, -3, Yes], [0, -1, 0, -2, Yes], [1, 1, 7, 5, Yes]], Yes)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([3, 2], [[-1, 0, 0, -3, Yes], [0, -1, 0, -2, Yes], [1, 1, 7, 5, Yes]], Yes)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0020","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af3c8d0b215f152c050acfe84543128a8400c9436806d97bf75df8a451db22c9","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216020,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,8]],\"point\":[4,3]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[4, 3], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 8]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([4, 3], [[-1, 0, 0, -4, Yes], [0, -1, 0, -3, Yes], [1, 1, 8, 7, Yes]], Yes)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([4, 3], [[-1, 0, 0, -4, Yes], [0, -1, 0, -3, Yes], [1, 1, 8, 7, Yes]], Yes)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0021","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acabc6dfa7ec79a34e3c40c293ef12d5b96c56c3b95187f219bdbcf187a3d56e","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216021,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,9]],\"point\":[0,0]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[0, 0], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 9]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([0, 0], [[-1, 0, 0, 0, Yes], [0, -1, 0, 0, Yes], [1, 1, 9, 0, Yes]], Yes)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([0, 0], [[-1, 0, 0, 0, Yes], [0, -1, 0, 0, Yes], [1, 1, 9, 0, Yes]], Yes)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0022","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74c209ed6133397fcceb529e5cf271f461c2ba277f9b921d1d2f868ba2f68f24","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216022,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,3]],\"point\":[1,1]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[1, 1], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 3]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([1, 1], [[-1, 0, 0, -1, Yes], [0, -1, 0, -1, Yes], [1, 1, 3, 2, Yes]], Yes)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([1, 1], [[-1, 0, 0, -1, Yes], [0, -1, 0, -1, Yes], [1, 1, 3, 2, Yes]], Yes)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0023","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af89bb15ccfc6421fc2ff451e35f0a2e719c934ab9d446cef2f3e7eee9032537","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216023,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,4]],\"point\":[2,2]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[2, 2], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 4]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([2, 2], [[-1, 0, 0, -2, Yes], [0, -1, 0, -2, Yes], [1, 1, 4, 4, Yes]], Yes)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([2, 2], [[-1, 0, 0, -2, Yes], [0, -1, 0, -2, Yes], [1, 1, 4, 4, Yes]], Yes)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0024","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08e71a0aa7c7a310ccc17222a790adb719b5fd7c2142d752462af8e2baaa8eca","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216024,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,5]],\"point\":[3,3]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[3, 3], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 5]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([3, 3], [[-1, 0, 0, -3, Yes], [0, -1, 0, -3, Yes], [1, 1, 5, 6, No]], No)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([3, 3], [[-1, 0, 0, -3, Yes], [0, -1, 0, -3, Yes], [1, 1, 5, 6, No]], No)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0025","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca6992cc8bc9fc52b9d4d33949cb16b1d0fbe238e004124e1c9bba41036805a1","family_id":"qf-app21-lp-constraint-row-audit","seed":202610216025,"semantic_template":"qf-app21-lp-constraint-row-audit/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,6]],\"point\":[4,0]}"},"fields":{"title":"Every Constraint at One Declared Point","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[4, 0], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 6]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?).","answer":{"kind":"expression","value":"([4, 0], [[-1, 0, 0, -4, Yes], [0, -1, 0, 0, Yes], [1, 1, 6, 4, Yes]], Yes)","explanation":"Check each inequality including equality at the boundary; infeasible candidates cannot be optimizers merely because their objective is better. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([4, 0], [[-1, 0, 0, -4, Yes], [0, -1, 0, 0, Yes], [1, 1, 6, 4, Yes]], Yes)."},"skills":["Every Constraint at One Declared Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0026","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12c2c9e8d9adc0e5bcc2ae1c342b7c174319eb9ce7ae4e27089bbf8bf9ce30f2","family_id":"qf-app21-lp-feasible-inventory","seed":202610216026,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,3]],\"points\":[[0,0],[0,2],[1,0],[2,0]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 0], [2, 0], [0, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 3]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 3]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 0, 2, [Yes, Yes, Yes], Yes, null], [2, 1, 0, [Yes, Yes, Yes], Yes, null], [3, 2, 0, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 3]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 0, 2, [Yes, Yes, Yes], Yes, null], [2, 1, 0, [Yes, Yes, Yes], Yes, null], [3, 2, 0, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0027","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecc12118c005744da83aacf2d9bb9903886b3a3706e6b382503c981748389e53","family_id":"qf-app21-lp-feasible-inventory","seed":202610216027,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,4]],\"points\":[[0,0],[1,1],[1,2],[2,1]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 1], [2, 1], [1, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 4]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 4]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 1, [Yes, Yes, Yes], Yes, null], [2, 1, 2, [Yes, Yes, Yes], Yes, null], [3, 2, 1, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 4]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 1, [Yes, Yes, Yes], Yes, null], [2, 1, 2, [Yes, Yes, Yes], Yes, null], [3, 2, 1, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0028","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1755bee90401f376edf4520cd5b646bb6b5f9bff8268d284a36d38bf7280459","family_id":"qf-app21-lp-feasible-inventory","seed":202610216028,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,5]],\"points\":[[0,0],[1,2],[2,2]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 2], [2, 2], [2, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 5]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 5]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 2, [Yes, Yes, Yes], Yes, null], [2, 2, 2, [Yes, Yes, Yes], Yes, null]], [0, 1, 2])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 5]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 2, [Yes, Yes, Yes], Yes, null], [2, 2, 2, [Yes, Yes, Yes], Yes, null]], [0, 1, 2])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0029","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e6fe2ef7959d7073e0deb0d52d8afaa629b3f97de79fdb07579b6cbf836e946","family_id":"qf-app21-lp-feasible-inventory","seed":202610216029,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,6]],\"points\":[[0,0],[1,3],[2,3],[3,2]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 3], [2, 3], [3, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 6]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 6]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 3, [Yes, Yes, Yes], Yes, null], [2, 2, 3, [Yes, Yes, Yes], Yes, null], [3, 3, 2, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 6]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 3, [Yes, Yes, Yes], Yes, null], [2, 2, 3, [Yes, Yes, Yes], Yes, null], [3, 3, 2, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0030","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3a7a20d4e38968deb550ac1f6aa2c4129378310c06f2fc7e9672b8352b89caa","family_id":"qf-app21-lp-feasible-inventory","seed":202610216030,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,7]],\"points\":[[0,0],[1,0],[2,4],[4,2]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 0], [2, 4], [4, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 7]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 7]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 0, [Yes, Yes, Yes], Yes, null], [2, 2, 4, [Yes, Yes, Yes], Yes, null], [3, 4, 2, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 7]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 0, [Yes, Yes, Yes], Yes, null], [2, 2, 4, [Yes, Yes, Yes], Yes, null], [3, 4, 2, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0031","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e77d80dd0393ea2996f66285df2e164efb6d30366348900d62afbcca0d96aac","family_id":"qf-app21-lp-feasible-inventory","seed":202610216031,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,8]],\"points\":[[0,0],[1,1],[2,0],[5,2]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 1], [2, 0], [5, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 8]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 8]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 1, [Yes, Yes, Yes], Yes, null], [2, 2, 0, [Yes, Yes, Yes], Yes, null], [3, 5, 2, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 8]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 1, [Yes, Yes, Yes], Yes, null], [2, 2, 0, [Yes, Yes, Yes], Yes, null], [3, 5, 2, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0032","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16d3c16ef84012e3ee9eda475f1e9dd2254c17d1763ab588fa619acfa6391be4","family_id":"qf-app21-lp-feasible-inventory","seed":202610216032,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,9]],\"points\":[[0,0],[0,2],[1,2],[2,1]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 2], [2, 1], [0, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 9]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 9]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 0, 2, [Yes, Yes, Yes], Yes, null], [2, 1, 2, [Yes, Yes, Yes], Yes, null], [3, 2, 1, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 9]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 0, 2, [Yes, Yes, Yes], Yes, null], [2, 1, 2, [Yes, Yes, Yes], Yes, null], [3, 2, 1, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0033","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05c7c11209ffe88f1dc95372d00ec43e4e2de3157c9ebb48ee3a29452d97da2c","family_id":"qf-app21-lp-feasible-inventory","seed":202610216033,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,3]],\"points\":[[0,0],[1,2],[1,3],[2,2]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 3], [2, 2], [1, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 3]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 3]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 2, [Yes, Yes, Yes], Yes, null], [2, 1, 3, [Yes, Yes, No], No, null], [3, 2, 2, [Yes, Yes, No], No, null]], [0, 1])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 3]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 2, [Yes, Yes, Yes], Yes, null], [2, 1, 3, [Yes, Yes, No], No, null], [3, 2, 2, [Yes, Yes, No], No, null]], [0, 1])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0034","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"533341e9eed39d9c7e3ad3f4684886fb56883d702d03c311c18aeefb42ea6402","family_id":"qf-app21-lp-feasible-inventory","seed":202610216034,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,4]],\"points\":[[0,0],[1,0],[2,2],[2,3]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 0], [2, 3], [2, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 4]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 4]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 0, [Yes, Yes, Yes], Yes, null], [2, 2, 2, [Yes, Yes, Yes], Yes, null], [3, 2, 3, [Yes, Yes, No], No, null]], [0, 1, 2])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 4]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 0, [Yes, Yes, Yes], Yes, null], [2, 2, 2, [Yes, Yes, Yes], Yes, null], [3, 2, 3, [Yes, Yes, No], No, null]], [0, 1, 2])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0035","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87515ff57ea1cfc9ab4c282c6088c54af0270cb0998d5c1be727ed070367b19f","family_id":"qf-app21-lp-feasible-inventory","seed":202610216035,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,5]],\"points\":[[0,0],[1,1],[2,4],[3,2]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 1], [2, 4], [3, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 5]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 5]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 1, [Yes, Yes, Yes], Yes, null], [2, 2, 4, [Yes, Yes, No], No, null], [3, 3, 2, [Yes, Yes, Yes], Yes, null]], [0, 1, 3])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 5]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 1, [Yes, Yes, Yes], Yes, null], [2, 2, 4, [Yes, Yes, No], No, null], [3, 3, 2, [Yes, Yes, Yes], Yes, null]], [0, 1, 3])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0036","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f39e28a501cdd272910cd99d5786f03d836eab27f00277c6b01e5401a2791f3","family_id":"qf-app21-lp-feasible-inventory","seed":202610216036,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,6]],\"points\":[[0,0],[1,2],[2,0],[4,2]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 2], [2, 0], [4, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 6]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 6]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 2, [Yes, Yes, Yes], Yes, null], [2, 2, 0, [Yes, Yes, Yes], Yes, null], [3, 4, 2, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 6]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 2, [Yes, Yes, Yes], Yes, null], [2, 2, 0, [Yes, Yes, Yes], Yes, null], [3, 4, 2, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0037","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e86c2c5523cc83d7ea7c04c167c91b0221f107c1f290832c078c2861028ff80","family_id":"qf-app21-lp-feasible-inventory","seed":202610216037,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,7]],\"points\":[[0,0],[1,3],[2,1],[5,2]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 3], [2, 1], [5, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 7]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 7]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 3, [Yes, Yes, Yes], Yes, null], [2, 2, 1, [Yes, Yes, Yes], Yes, null], [3, 5, 2, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 7]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 3, [Yes, Yes, Yes], Yes, null], [2, 2, 1, [Yes, Yes, Yes], Yes, null], [3, 5, 2, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0038","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"966b017ee9ca238532e7c2fcfd0062856b5be8d701d3969856159cbc6013fe84","family_id":"qf-app21-lp-feasible-inventory","seed":202610216038,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,8]],\"points\":[[0,0],[0,2],[1,0],[2,2]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 0], [2, 2], [0, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 8]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 8]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 0, 2, [Yes, Yes, Yes], Yes, null], [2, 1, 0, [Yes, Yes, Yes], Yes, null], [3, 2, 2, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 8]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 0, 2, [Yes, Yes, Yes], Yes, null], [2, 1, 0, [Yes, Yes, Yes], Yes, null], [3, 2, 2, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0039","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab4245a9f412249ee5a740e06e29a0ed27cc8144285956075d2fcc10427ad191","family_id":"qf-app21-lp-feasible-inventory","seed":202610216039,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,9]],\"points\":[[0,0],[1,1],[1,2],[2,3]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 1], [2, 3], [1, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 9]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 9]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 1, [Yes, Yes, Yes], Yes, null], [2, 1, 2, [Yes, Yes, Yes], Yes, null], [3, 2, 3, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 9]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 1, [Yes, Yes, Yes], Yes, null], [2, 1, 2, [Yes, Yes, Yes], Yes, null], [3, 2, 3, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0040","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7e2d59e9a30e9bab933b7022c14058f9fef4c69ad2331b12a29acb12663affc","family_id":"qf-app21-lp-feasible-inventory","seed":202610216040,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,3]],\"points\":[[0,0],[1,2],[2,2],[2,4]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 2], [2, 4], [2, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 3]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 3]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 2, [Yes, Yes, Yes], Yes, null], [2, 2, 2, [Yes, Yes, No], No, null], [3, 2, 4, [Yes, Yes, No], No, null]], [0, 1])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 3]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 2, [Yes, Yes, Yes], Yes, null], [2, 2, 2, [Yes, Yes, No], No, null], [3, 2, 4, [Yes, Yes, No], No, null]], [0, 1])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0041","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"719c66ff140f732deb80dd754b7137cd31ee3e44b2337c0ce4c44a776c17b087","family_id":"qf-app21-lp-feasible-inventory","seed":202610216041,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,4]],\"points\":[[0,0],[1,3],[2,0],[3,2]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 3], [2, 0], [3, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 4]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 4]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 3, [Yes, Yes, Yes], Yes, null], [2, 2, 0, [Yes, Yes, Yes], Yes, null], [3, 3, 2, [Yes, Yes, No], No, null]], [0, 1, 2])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 4]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 3, [Yes, Yes, Yes], Yes, null], [2, 2, 0, [Yes, Yes, Yes], Yes, null], [3, 3, 2, [Yes, Yes, No], No, null]], [0, 1, 2])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0042","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fd65c8047337e9ca58507ac04db5a686c4a3b02d43f1c0d372638dd95ae70d0","family_id":"qf-app21-lp-feasible-inventory","seed":202610216042,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,5]],\"points\":[[0,0],[1,0],[2,1],[4,2]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 0], [2, 1], [4, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 5]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 5]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 0, [Yes, Yes, Yes], Yes, null], [2, 2, 1, [Yes, Yes, Yes], Yes, null], [3, 4, 2, [Yes, Yes, No], No, null]], [0, 1, 2])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 5]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 0, [Yes, Yes, Yes], Yes, null], [2, 2, 1, [Yes, Yes, Yes], Yes, null], [3, 4, 2, [Yes, Yes, No], No, null]], [0, 1, 2])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0043","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0760c69fbf1d220c62b4218ae641c87e1ebbd65b0280c870ce7716e7c1e62872","family_id":"qf-app21-lp-feasible-inventory","seed":202610216043,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,6]],\"points\":[[0,0],[1,1],[2,2],[5,2]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 1], [2, 2], [5, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 6]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 6]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 1, [Yes, Yes, Yes], Yes, null], [2, 2, 2, [Yes, Yes, Yes], Yes, null], [3, 5, 2, [Yes, Yes, No], No, null]], [0, 1, 2])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 6]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 1, [Yes, Yes, Yes], Yes, null], [2, 2, 2, [Yes, Yes, Yes], Yes, null], [3, 5, 2, [Yes, Yes, No], No, null]], [0, 1, 2])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0044","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83563eb91c5ecb00f2ff5676c0c192f38bc887d523ea7e2347988a3e13d7677d","family_id":"qf-app21-lp-feasible-inventory","seed":202610216044,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,7]],\"points\":[[0,0],[0,2],[1,2],[2,3]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 2], [2, 3], [0, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 7]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 7]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 0, 2, [Yes, Yes, Yes], Yes, null], [2, 1, 2, [Yes, Yes, Yes], Yes, null], [3, 2, 3, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 7]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 0, 2, [Yes, Yes, Yes], Yes, null], [2, 1, 2, [Yes, Yes, Yes], Yes, null], [3, 2, 3, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0045","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a5f7aa5df612d19b090896cb6b49da5e1c719b352162f78df3ae4b1c70c76a1","family_id":"qf-app21-lp-feasible-inventory","seed":202610216045,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,8]],\"points\":[[0,0],[1,2],[1,3],[2,4]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 3], [2, 4], [1, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 8]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 8]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 2, [Yes, Yes, Yes], Yes, null], [2, 1, 3, [Yes, Yes, Yes], Yes, null], [3, 2, 4, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 8]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 2, [Yes, Yes, Yes], Yes, null], [2, 1, 3, [Yes, Yes, Yes], Yes, null], [3, 2, 4, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0046","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c708a8655d9d50bf0591dfffbf6a63b0492e3c7827d9e1ed6a0a798fa8db56e7","family_id":"qf-app21-lp-feasible-inventory","seed":202610216046,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,9]],\"points\":[[0,0],[1,0],[2,0],[2,2]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 0], [2, 0], [2, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 9]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 9]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 0, [Yes, Yes, Yes], Yes, null], [2, 2, 0, [Yes, Yes, Yes], Yes, null], [3, 2, 2, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 9]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 0, [Yes, Yes, Yes], Yes, null], [2, 2, 0, [Yes, Yes, Yes], Yes, null], [3, 2, 2, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0047","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cc5699a7678411fbdeb97761035caf5a3dd721d2eadd382e04a7a21f5bbac6d","family_id":"qf-app21-lp-feasible-inventory","seed":202610216047,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,3]],\"points\":[[0,0],[1,1],[2,1],[3,2]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 1], [2, 1], [3, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 3]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 3]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 1, [Yes, Yes, Yes], Yes, null], [2, 2, 1, [Yes, Yes, Yes], Yes, null], [3, 3, 2, [Yes, Yes, No], No, null]], [0, 1, 2])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 3]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 1, [Yes, Yes, Yes], Yes, null], [2, 2, 1, [Yes, Yes, Yes], Yes, null], [3, 3, 2, [Yes, Yes, No], No, null]], [0, 1, 2])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0048","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78fecc369480a93ab1d17d3aab7f81f66b5a9cb0446b44cc00744ce259422a44","family_id":"qf-app21-lp-feasible-inventory","seed":202610216048,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,4]],\"points\":[[0,0],[1,2],[2,2],[4,2]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 2], [2, 2], [4, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 4]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 4]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 2, [Yes, Yes, Yes], Yes, null], [2, 2, 2, [Yes, Yes, Yes], Yes, null], [3, 4, 2, [Yes, Yes, No], No, null]], [0, 1, 2])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 4]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 2, [Yes, Yes, Yes], Yes, null], [2, 2, 2, [Yes, Yes, Yes], Yes, null], [3, 4, 2, [Yes, Yes, No], No, null]], [0, 1, 2])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0049","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89649f59363d633044b74a1780c21efe72eda17970a625df40fed369227cc5b6","family_id":"qf-app21-lp-feasible-inventory","seed":202610216049,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,5]],\"points\":[[0,0],[1,3],[2,3],[5,2]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 3], [2, 3], [5, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 5]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 5]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 3, [Yes, Yes, Yes], Yes, null], [2, 2, 3, [Yes, Yes, Yes], Yes, null], [3, 5, 2, [Yes, Yes, No], No, null]], [0, 1, 2])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 5]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 3, [Yes, Yes, Yes], Yes, null], [2, 2, 3, [Yes, Yes, Yes], Yes, null], [3, 5, 2, [Yes, Yes, No], No, null]], [0, 1, 2])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0050","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"408ec442daffb7ddd3e78a8391591bf184207377281f9521628ab8b7806b21de","family_id":"qf-app21-lp-feasible-inventory","seed":202610216050,"semantic_template":"qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,6]],\"points\":[[0,0],[0,2],[1,0],[2,4]]}"},"fields":{"title":"Feasible Indices in Canonical Candidate-Grid Order","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 0], [2, 4], [0, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 6]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 6]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 0, 2, [Yes, Yes, Yes], Yes, null], [2, 1, 0, [Yes, Yes, Yes], Yes, null], [3, 2, 4, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])","explanation":"Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 6]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 0, 2, [Yes, Yes, Yes], Yes, null], [2, 1, 0, [Yes, Yes, Yes], Yes, null], [3, 2, 4, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])."},"skills":["Feasible Indices in Canonical Candidate-Grid Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0051","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"097b50d233922d21117f8a2c289535ce5bb46079f972058f3a47879147a5477c","family_id":"qf-app21-lp-objective-audit","seed":202610216051,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[-2,2],\"points\":[[0,0],[0,2],[1,0],[2,0]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 0], [2, 0], [0, 2]], objective [u,v]=[-2, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([-2, 2], [[0, 0, 0, 0], [1, 0, 2, 4], [2, 1, 0, -2], [3, 2, 0, -4]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-2, 2], [[0, 0, 0, 0], [1, 0, 2, 4], [2, 1, 0, -2], [3, 2, 0, -4]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0052","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94e9f23a8ea72914334609830e077fdaed8bc817690637d29cc363bfb4d50520","family_id":"qf-app21-lp-objective-audit","seed":202610216052,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[-1,2],\"points\":[[0,0],[1,1],[1,2],[2,1]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 1], [2, 1], [1, 2]], objective [u,v]=[-1, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([-1, 2], [[0, 0, 0, 0], [1, 1, 1, 1], [2, 1, 2, 3], [3, 2, 1, 0]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-1, 2], [[0, 0, 0, 0], [1, 1, 1, 1], [2, 1, 2, 3], [3, 2, 1, 0]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0053","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecb856e9c744099f124b1dd87d40a7f152f103ce65ee1e093e4b6ec51f83d64e","family_id":"qf-app21-lp-objective-audit","seed":202610216053,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[0,2],\"points\":[[0,0],[1,2],[2,2]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 2], [2, 2], [2, 2]], objective [u,v]=[0, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([0, 2], [[0, 0, 0, 0], [1, 1, 2, 4], [2, 2, 2, 4]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([0, 2], [[0, 0, 0, 0], [1, 1, 2, 4], [2, 2, 2, 4]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0054","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d2f6a2caeb0b91eaaf1a438e9efc58e8daf75c6928f47530e5bcdf2d904d1b6","family_id":"qf-app21-lp-objective-audit","seed":202610216054,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[1,2],\"points\":[[0,0],[1,3],[2,3],[3,2]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 3], [2, 3], [3, 2]], objective [u,v]=[1, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([1, 2], [[0, 0, 0, 0], [1, 1, 3, 7], [2, 2, 3, 8], [3, 3, 2, 7]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([1, 2], [[0, 0, 0, 0], [1, 1, 3, 7], [2, 2, 3, 8], [3, 3, 2, 7]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0055","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c9f41a3ac70bbabecc9012e8303e6e695be62b38f2e452de120cd5181afde3d","family_id":"qf-app21-lp-objective-audit","seed":202610216055,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[2,2],\"points\":[[0,0],[1,0],[2,4],[4,2]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 0], [2, 4], [4, 2]], objective [u,v]=[2, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([2, 2], [[0, 0, 0, 0], [1, 1, 0, 2], [2, 2, 4, 12], [3, 4, 2, 12]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([2, 2], [[0, 0, 0, 0], [1, 1, 0, 2], [2, 2, 4, 12], [3, 4, 2, 12]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0056","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6191d3ac04aa2ca689b49ba91241a706a90881b3b3722c64211221501ad7fec9","family_id":"qf-app21-lp-objective-audit","seed":202610216056,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[-2,2],\"points\":[[0,0],[1,1],[2,0],[5,2]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 1], [2, 0], [5, 2]], objective [u,v]=[-2, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([-2, 2], [[0, 0, 0, 0], [1, 1, 1, 0], [2, 2, 0, -4], [3, 5, 2, -6]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-2, 2], [[0, 0, 0, 0], [1, 1, 1, 0], [2, 2, 0, -4], [3, 5, 2, -6]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0057","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16ccdbe6a990b7ef675c046917baacb90077e6dac67b1f8a8760dccb5c294f4a","family_id":"qf-app21-lp-objective-audit","seed":202610216057,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[-1,2],\"points\":[[0,0],[0,2],[1,2],[2,1]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 2], [2, 1], [0, 2]], objective [u,v]=[-1, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([-1, 2], [[0, 0, 0, 0], [1, 0, 2, 4], [2, 1, 2, 3], [3, 2, 1, 0]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-1, 2], [[0, 0, 0, 0], [1, 0, 2, 4], [2, 1, 2, 3], [3, 2, 1, 0]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0058","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84b676bb108e29dfadfbed015b9773d0c4e4e1275c060791b47fa71ca50e7cd6","family_id":"qf-app21-lp-objective-audit","seed":202610216058,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[0,2],\"points\":[[0,0],[1,2],[1,3],[2,2]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 3], [2, 2], [1, 2]], objective [u,v]=[0, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([0, 2], [[0, 0, 0, 0], [1, 1, 2, 4], [2, 1, 3, 6], [3, 2, 2, 4]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([0, 2], [[0, 0, 0, 0], [1, 1, 2, 4], [2, 1, 3, 6], [3, 2, 2, 4]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0059","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc29563675419b696d1202169c8f796a2f9da0a27d41565eabdcf6a8122a7d65","family_id":"qf-app21-lp-objective-audit","seed":202610216059,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[1,2],\"points\":[[0,0],[1,0],[2,2],[2,3]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 0], [2, 3], [2, 2]], objective [u,v]=[1, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([1, 2], [[0, 0, 0, 0], [1, 1, 0, 1], [2, 2, 2, 6], [3, 2, 3, 8]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([1, 2], [[0, 0, 0, 0], [1, 1, 0, 1], [2, 2, 2, 6], [3, 2, 3, 8]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0060","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c99bd43547f14fdd774d6d12a8ec380dffd0161746a2e1300f20400b5e9467a7","family_id":"qf-app21-lp-objective-audit","seed":202610216060,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[2,2],\"points\":[[0,0],[1,1],[2,4],[3,2]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 1], [2, 4], [3, 2]], objective [u,v]=[2, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([2, 2], [[0, 0, 0, 0], [1, 1, 1, 4], [2, 2, 4, 12], [3, 3, 2, 10]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([2, 2], [[0, 0, 0, 0], [1, 1, 1, 4], [2, 2, 4, 12], [3, 3, 2, 10]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0061","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3af433154e21e870c7b22acc9e58adf18653426d5da180b423d7af44846ad82d","family_id":"qf-app21-lp-objective-audit","seed":202610216061,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[-2,2],\"points\":[[0,0],[1,2],[2,0],[4,2]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 2], [2, 0], [4, 2]], objective [u,v]=[-2, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([-2, 2], [[0, 0, 0, 0], [1, 1, 2, 2], [2, 2, 0, -4], [3, 4, 2, -4]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-2, 2], [[0, 0, 0, 0], [1, 1, 2, 2], [2, 2, 0, -4], [3, 4, 2, -4]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0062","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cf7e0945bee02ae4cfc0dad92839c84d8ae69ac0b0dd4150863859b8382687f","family_id":"qf-app21-lp-objective-audit","seed":202610216062,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[-1,2],\"points\":[[0,0],[1,3],[2,1],[5,2]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 3], [2, 1], [5, 2]], objective [u,v]=[-1, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([-1, 2], [[0, 0, 0, 0], [1, 1, 3, 5], [2, 2, 1, 0], [3, 5, 2, -1]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-1, 2], [[0, 0, 0, 0], [1, 1, 3, 5], [2, 2, 1, 0], [3, 5, 2, -1]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0063","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c634fb585d0a7da44f71567dea7a9effd244b4d1e44e16640e529f202d522c4","family_id":"qf-app21-lp-objective-audit","seed":202610216063,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[0,2],\"points\":[[0,0],[0,2],[1,0],[2,2]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 0], [2, 2], [0, 2]], objective [u,v]=[0, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([0, 2], [[0, 0, 0, 0], [1, 0, 2, 4], [2, 1, 0, 0], [3, 2, 2, 4]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([0, 2], [[0, 0, 0, 0], [1, 0, 2, 4], [2, 1, 0, 0], [3, 2, 2, 4]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0064","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4b09b319f17ec2ee2d7e1a4d23ddee1a09ee515bd195f53a95d26d807f2bc81","family_id":"qf-app21-lp-objective-audit","seed":202610216064,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[1,2],\"points\":[[0,0],[1,1],[1,2],[2,3]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 1], [2, 3], [1, 2]], objective [u,v]=[1, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([1, 2], [[0, 0, 0, 0], [1, 1, 1, 3], [2, 1, 2, 5], [3, 2, 3, 8]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([1, 2], [[0, 0, 0, 0], [1, 1, 1, 3], [2, 1, 2, 5], [3, 2, 3, 8]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0065","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fd3de81d09ae824b994a71ebef42d6b3c279ed5477b8a93b48cc2a7748b7d23","family_id":"qf-app21-lp-objective-audit","seed":202610216065,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[2,2],\"points\":[[0,0],[1,2],[2,2],[2,4]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 2], [2, 4], [2, 2]], objective [u,v]=[2, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([2, 2], [[0, 0, 0, 0], [1, 1, 2, 6], [2, 2, 2, 8], [3, 2, 4, 12]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([2, 2], [[0, 0, 0, 0], [1, 1, 2, 6], [2, 2, 2, 8], [3, 2, 4, 12]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0066","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e0c0a5e62f723eb1844e3e9b8115b8a46b0d3cdc846cb345d8dadd4c611ea84","family_id":"qf-app21-lp-objective-audit","seed":202610216066,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[-2,2],\"points\":[[0,0],[1,3],[2,0],[3,2]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 3], [2, 0], [3, 2]], objective [u,v]=[-2, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([-2, 2], [[0, 0, 0, 0], [1, 1, 3, 4], [2, 2, 0, -4], [3, 3, 2, -2]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-2, 2], [[0, 0, 0, 0], [1, 1, 3, 4], [2, 2, 0, -4], [3, 3, 2, -2]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0067","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ff044cd91aa70de54677e84c4b63998ad546e8837e38a20c78c120202963d31","family_id":"qf-app21-lp-objective-audit","seed":202610216067,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[-1,2],\"points\":[[0,0],[1,0],[2,1],[4,2]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 0], [2, 1], [4, 2]], objective [u,v]=[-1, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([-1, 2], [[0, 0, 0, 0], [1, 1, 0, -1], [2, 2, 1, 0], [3, 4, 2, 0]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-1, 2], [[0, 0, 0, 0], [1, 1, 0, -1], [2, 2, 1, 0], [3, 4, 2, 0]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0068","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5915dda8b429be307076ca723aba40b218be39852c58a020587815124f79e2c","family_id":"qf-app21-lp-objective-audit","seed":202610216068,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[0,2],\"points\":[[0,0],[1,1],[2,2],[5,2]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 1], [2, 2], [5, 2]], objective [u,v]=[0, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([0, 2], [[0, 0, 0, 0], [1, 1, 1, 2], [2, 2, 2, 4], [3, 5, 2, 4]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([0, 2], [[0, 0, 0, 0], [1, 1, 1, 2], [2, 2, 2, 4], [3, 5, 2, 4]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0069","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc0e55700b4cff30e0d11f23f5c447d9224b5822811a447b44b976d53c36d3e3","family_id":"qf-app21-lp-objective-audit","seed":202610216069,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[1,2],\"points\":[[0,0],[0,2],[1,2],[2,3]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 2], [2, 3], [0, 2]], objective [u,v]=[1, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([1, 2], [[0, 0, 0, 0], [1, 0, 2, 4], [2, 1, 2, 5], [3, 2, 3, 8]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([1, 2], [[0, 0, 0, 0], [1, 0, 2, 4], [2, 1, 2, 5], [3, 2, 3, 8]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0070","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d532d7d7e9af5cc3234ad7e47a99218436d18e0dcd02a87a59012ba61e2fa57","family_id":"qf-app21-lp-objective-audit","seed":202610216070,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[2,2],\"points\":[[0,0],[1,2],[1,3],[2,4]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 3], [2, 4], [1, 2]], objective [u,v]=[2, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([2, 2], [[0, 0, 0, 0], [1, 1, 2, 6], [2, 1, 3, 8], [3, 2, 4, 12]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([2, 2], [[0, 0, 0, 0], [1, 1, 2, 6], [2, 1, 3, 8], [3, 2, 4, 12]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0071","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68b9f0f962db3832cda82165bbf78496521de4b3697a62d96a402245178dc670","family_id":"qf-app21-lp-objective-audit","seed":202610216071,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[-2,2],\"points\":[[0,0],[1,0],[2,0],[2,2]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 0], [2, 0], [2, 2]], objective [u,v]=[-2, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([-2, 2], [[0, 0, 0, 0], [1, 1, 0, -2], [2, 2, 0, -4], [3, 2, 2, 0]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-2, 2], [[0, 0, 0, 0], [1, 1, 0, -2], [2, 2, 0, -4], [3, 2, 2, 0]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0072","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e501604ee9ba8f623f80a710b12eb8f8fd8c886508eb20fab730897573304f3","family_id":"qf-app21-lp-objective-audit","seed":202610216072,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[-1,2],\"points\":[[0,0],[1,1],[2,1],[3,2]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 1], [2, 1], [3, 2]], objective [u,v]=[-1, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([-1, 2], [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 1, 0], [3, 3, 2, 1]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-1, 2], [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 1, 0], [3, 3, 2, 1]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0073","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82c892672a4dc66c76ed71c07309532438055c4b48bc6f08ca08a431345f8200","family_id":"qf-app21-lp-objective-audit","seed":202610216073,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[0,2],\"points\":[[0,0],[1,2],[2,2],[4,2]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 2], [2, 2], [4, 2]], objective [u,v]=[0, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([0, 2], [[0, 0, 0, 0], [1, 1, 2, 4], [2, 2, 2, 4], [3, 4, 2, 4]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([0, 2], [[0, 0, 0, 0], [1, 1, 2, 4], [2, 2, 2, 4], [3, 4, 2, 4]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0074","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cadda6455d7aabfc5bf92f3e502ed467721cbdcfe79292e285ecb166949192fb","family_id":"qf-app21-lp-objective-audit","seed":202610216074,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[1,2],\"points\":[[0,0],[1,3],[2,3],[5,2]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 3], [2, 3], [5, 2]], objective [u,v]=[1, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([1, 2], [[0, 0, 0, 0], [1, 1, 3, 7], [2, 2, 3, 8], [3, 5, 2, 9]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([1, 2], [[0, 0, 0, 0], [1, 1, 3, 7], [2, 2, 3, 8], [3, 5, 2, 9]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0075","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be23a91d13c1a0e9ca5e56aad46b3cab6332ac8fea928c37c1beaa03f680ccca","family_id":"qf-app21-lp-objective-audit","seed":202610216075,"semantic_template":"qf-app21-lp-objective-audit/{\"objective\":[2,2],\"points\":[[0,0],[0,2],[1,0],[2,4]]}"},"fields":{"title":"A Linear Objective Does Not by Itself Establish Feasibility","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 0], [2, 4], [0, 2]], objective [u,v]=[2, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.","answer":{"kind":"expression","value":"([2, 2], [[0, 0, 0, 0], [1, 0, 2, 4], [2, 1, 0, 2], [3, 2, 4, 12]])","explanation":"Evaluate the linear form at each distinct canonical point; do not invent feasibility or an optimum when the requested task is only objective evaluation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([2, 2], [[0, 0, 0, 0], [1, 0, 2, 4], [2, 1, 0, 2], [3, 2, 4, 12]])."},"skills":["A Linear Objective Does Not by Itself Establish Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0076","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72951a309add7f2f4c92909329956b2e82e3df02c0de4b67747f369a643841f5","family_id":"qf-app21-lp-finite-optimum","seed":202610216076,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,3]],\"direction\":\"max\",\"objective\":[-2,2],\"points\":[[0,0],[0,2],[1,0],[2,0]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 0], [2, 0], [0, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 3]], objective=[-2, 2], direction=max. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 3]], [-2, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 0, 2, [Yes, Yes, Yes], Yes, 4], [2, 1, 0, [Yes, Yes, Yes], Yes, -2], [3, 2, 0, [Yes, Yes, Yes], Yes, -4]], 4, [1])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 3]], [-2, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 0, 2, [Yes, Yes, Yes], Yes, 4], [2, 1, 0, [Yes, Yes, Yes], Yes, -2], [3, 2, 0, [Yes, Yes, Yes], Yes, -4]], 4, [1])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0077","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19feb86a9da9873fa98a7e1815acc17addcdb9fe5e5a90e3e90d81e608a5dd27","family_id":"qf-app21-lp-finite-optimum","seed":202610216077,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,4]],\"direction\":\"min\",\"objective\":[-1,2],\"points\":[[0,0],[1,1],[1,2],[2,1]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 1], [2, 1], [1, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 4]], objective=[-1, 2], direction=min. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 4]], [-1, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 1, [Yes, Yes, Yes], Yes, 1], [2, 1, 2, [Yes, Yes, Yes], Yes, 3], [3, 2, 1, [Yes, Yes, Yes], Yes, 0]], 0, [0, 3])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 4]], [-1, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 1, [Yes, Yes, Yes], Yes, 1], [2, 1, 2, [Yes, Yes, Yes], Yes, 3], [3, 2, 1, [Yes, Yes, Yes], Yes, 0]], 0, [0, 3])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0078","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe138ef964f38332c3c59b96c963e2aa722caab72df4a09b69305375939b0723","family_id":"qf-app21-lp-finite-optimum","seed":202610216078,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,5]],\"direction\":\"max\",\"objective\":[0,2],\"points\":[[0,0],[1,2],[2,2]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 2], [2, 2], [2, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 5]], objective=[0, 2], direction=max. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 5]], [0, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 2, [Yes, Yes, Yes], Yes, 4], [2, 2, 2, [Yes, Yes, Yes], Yes, 4]], 4, [1, 2])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 5]], [0, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 2, [Yes, Yes, Yes], Yes, 4], [2, 2, 2, [Yes, Yes, Yes], Yes, 4]], 4, [1, 2])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0079","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d11f5bcfca0a00d70ec54f516c31dc7bd7bbc14b820ae7a5ebef17ea1fc3c083","family_id":"qf-app21-lp-finite-optimum","seed":202610216079,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,6]],\"direction\":\"min\",\"objective\":[1,2],\"points\":[[0,0],[1,3],[2,3],[3,2]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 3], [2, 3], [3, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 6]], objective=[1, 2], direction=min. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 6]], [1, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 3, [Yes, Yes, Yes], Yes, 7], [2, 2, 3, [Yes, Yes, Yes], Yes, 8], [3, 3, 2, [Yes, Yes, Yes], Yes, 7]], 0, [0])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 6]], [1, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 3, [Yes, Yes, Yes], Yes, 7], [2, 2, 3, [Yes, Yes, Yes], Yes, 8], [3, 3, 2, [Yes, Yes, Yes], Yes, 7]], 0, [0])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0080","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d46df97f36f77f4561eead01b6fd0227e11462376c03e8d920cf4cd038097dde","family_id":"qf-app21-lp-finite-optimum","seed":202610216080,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,7]],\"direction\":\"max\",\"objective\":[2,2],\"points\":[[0,0],[1,0],[2,4],[4,2]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 0], [2, 4], [4, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 7]], objective=[2, 2], direction=max. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 7]], [2, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 0, [Yes, Yes, Yes], Yes, 2], [2, 2, 4, [Yes, Yes, Yes], Yes, 12], [3, 4, 2, [Yes, Yes, Yes], Yes, 12]], 12, [2, 3])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 7]], [2, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 0, [Yes, Yes, Yes], Yes, 2], [2, 2, 4, [Yes, Yes, Yes], Yes, 12], [3, 4, 2, [Yes, Yes, Yes], Yes, 12]], 12, [2, 3])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0081","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d3e99884b65bcb9b6adb126cc8b0c80bb7da9b3e5bf6cac74595b3203c4e8fe","family_id":"qf-app21-lp-finite-optimum","seed":202610216081,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,8]],\"direction\":\"min\",\"objective\":[-2,2],\"points\":[[0,0],[1,1],[2,0],[5,2]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 1], [2, 0], [5, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 8]], objective=[-2, 2], direction=min. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 8]], [-2, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 1, [Yes, Yes, Yes], Yes, 0], [2, 2, 0, [Yes, Yes, Yes], Yes, -4], [3, 5, 2, [Yes, Yes, Yes], Yes, -6]], -6, [3])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 8]], [-2, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 1, [Yes, Yes, Yes], Yes, 0], [2, 2, 0, [Yes, Yes, Yes], Yes, -4], [3, 5, 2, [Yes, Yes, Yes], Yes, -6]], -6, [3])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0082","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7af8b42bea38e7871b462399cd07a076faabfb6832701033e1000f56a2c113fd","family_id":"qf-app21-lp-finite-optimum","seed":202610216082,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,9]],\"direction\":\"max\",\"objective\":[-1,2],\"points\":[[0,0],[0,2],[1,2],[2,1]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 2], [2, 1], [0, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 9]], objective=[-1, 2], direction=max. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 9]], [-1, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 0, 2, [Yes, Yes, Yes], Yes, 4], [2, 1, 2, [Yes, Yes, Yes], Yes, 3], [3, 2, 1, [Yes, Yes, Yes], Yes, 0]], 4, [1])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 9]], [-1, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 0, 2, [Yes, Yes, Yes], Yes, 4], [2, 1, 2, [Yes, Yes, Yes], Yes, 3], [3, 2, 1, [Yes, Yes, Yes], Yes, 0]], 4, [1])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0083","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"018731bcea6dc8e79ef35fccd8391105235580516b73f538591b80e32147214d","family_id":"qf-app21-lp-finite-optimum","seed":202610216083,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,3]],\"direction\":\"min\",\"objective\":[0,2],\"points\":[[0,0],[1,2],[1,3],[2,2]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 3], [2, 2], [1, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 3]], objective=[0, 2], direction=min. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 3]], [0, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 2, [Yes, Yes, Yes], Yes, 4], [2, 1, 3, [Yes, Yes, No], No, 6], [3, 2, 2, [Yes, Yes, No], No, 4]], 0, [0])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 3]], [0, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 2, [Yes, Yes, Yes], Yes, 4], [2, 1, 3, [Yes, Yes, No], No, 6], [3, 2, 2, [Yes, Yes, No], No, 4]], 0, [0])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0084","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1b43a56046949144ee041fdccb5354f5b908e7d781285e0b363195450bf6281","family_id":"qf-app21-lp-finite-optimum","seed":202610216084,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,4]],\"direction\":\"max\",\"objective\":[1,2],\"points\":[[0,0],[1,0],[2,2],[2,3]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 0], [2, 3], [2, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 4]], objective=[1, 2], direction=max. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 4]], [1, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 0, [Yes, Yes, Yes], Yes, 1], [2, 2, 2, [Yes, Yes, Yes], Yes, 6], [3, 2, 3, [Yes, Yes, No], No, 8]], 6, [2])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 4]], [1, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 0, [Yes, Yes, Yes], Yes, 1], [2, 2, 2, [Yes, Yes, Yes], Yes, 6], [3, 2, 3, [Yes, Yes, No], No, 8]], 6, [2])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0085","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d419550ca1047b27faf1251ac8a23ffa043cf24ec872e1c23acf27e092c612a","family_id":"qf-app21-lp-finite-optimum","seed":202610216085,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,5]],\"direction\":\"min\",\"objective\":[2,2],\"points\":[[0,0],[1,1],[2,4],[3,2]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 1], [2, 4], [3, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 5]], objective=[2, 2], direction=min. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 5]], [2, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 1, [Yes, Yes, Yes], Yes, 4], [2, 2, 4, [Yes, Yes, No], No, 12], [3, 3, 2, [Yes, Yes, Yes], Yes, 10]], 0, [0])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 5]], [2, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 1, [Yes, Yes, Yes], Yes, 4], [2, 2, 4, [Yes, Yes, No], No, 12], [3, 3, 2, [Yes, Yes, Yes], Yes, 10]], 0, [0])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0086","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7393a4b4faec3af7efe921595be616da783a028136b2c8d5282faa4f8de8e76f","family_id":"qf-app21-lp-finite-optimum","seed":202610216086,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,6]],\"direction\":\"max\",\"objective\":[-2,2],\"points\":[[0,0],[1,2],[2,0],[4,2]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 2], [2, 0], [4, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 6]], objective=[-2, 2], direction=max. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 6]], [-2, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 2, [Yes, Yes, Yes], Yes, 2], [2, 2, 0, [Yes, Yes, Yes], Yes, -4], [3, 4, 2, [Yes, Yes, Yes], Yes, -4]], 2, [1])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 6]], [-2, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 2, [Yes, Yes, Yes], Yes, 2], [2, 2, 0, [Yes, Yes, Yes], Yes, -4], [3, 4, 2, [Yes, Yes, Yes], Yes, -4]], 2, [1])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0087","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b43a553a8de03d07dca63224110dd3a99acc261e29db22e8078de5520766840d","family_id":"qf-app21-lp-finite-optimum","seed":202610216087,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,7]],\"direction\":\"min\",\"objective\":[-1,2],\"points\":[[0,0],[1,3],[2,1],[5,2]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 3], [2, 1], [5, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 7]], objective=[-1, 2], direction=min. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 7]], [-1, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 3, [Yes, Yes, Yes], Yes, 5], [2, 2, 1, [Yes, Yes, Yes], Yes, 0], [3, 5, 2, [Yes, Yes, Yes], Yes, -1]], -1, [3])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 7]], [-1, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 3, [Yes, Yes, Yes], Yes, 5], [2, 2, 1, [Yes, Yes, Yes], Yes, 0], [3, 5, 2, [Yes, Yes, Yes], Yes, -1]], -1, [3])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0088","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80e39953bc78e18ee17b53743ad9e6ef434eb8cd31203f3c149834532d6304f7","family_id":"qf-app21-lp-finite-optimum","seed":202610216088,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,8]],\"direction\":\"max\",\"objective\":[0,2],\"points\":[[0,0],[0,2],[1,0],[2,2]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 0], [2, 2], [0, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 8]], objective=[0, 2], direction=max. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 8]], [0, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 0, 2, [Yes, Yes, Yes], Yes, 4], [2, 1, 0, [Yes, Yes, Yes], Yes, 0], [3, 2, 2, [Yes, Yes, Yes], Yes, 4]], 4, [1, 3])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 8]], [0, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 0, 2, [Yes, Yes, Yes], Yes, 4], [2, 1, 0, [Yes, Yes, Yes], Yes, 0], [3, 2, 2, [Yes, Yes, Yes], Yes, 4]], 4, [1, 3])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0089","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67f86a5881e8324408fef27ee8bcea7e4ea3a9a5c249f910e5c78cca0002833f","family_id":"qf-app21-lp-finite-optimum","seed":202610216089,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,9]],\"direction\":\"min\",\"objective\":[1,2],\"points\":[[0,0],[1,1],[1,2],[2,3]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 1], [2, 3], [1, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 9]], objective=[1, 2], direction=min. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 9]], [1, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 1, [Yes, Yes, Yes], Yes, 3], [2, 1, 2, [Yes, Yes, Yes], Yes, 5], [3, 2, 3, [Yes, Yes, Yes], Yes, 8]], 0, [0])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 9]], [1, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 1, [Yes, Yes, Yes], Yes, 3], [2, 1, 2, [Yes, Yes, Yes], Yes, 5], [3, 2, 3, [Yes, Yes, Yes], Yes, 8]], 0, [0])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0090","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bde63d9fd0d1b73798e0cac0ba49c6ad6e228518aaa62003cea7729e58b2f07a","family_id":"qf-app21-lp-finite-optimum","seed":202610216090,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,3]],\"direction\":\"max\",\"objective\":[2,2],\"points\":[[0,0],[1,2],[2,2],[2,4]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 2], [2, 4], [2, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 3]], objective=[2, 2], direction=max. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 3]], [2, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 2, [Yes, Yes, Yes], Yes, 6], [2, 2, 2, [Yes, Yes, No], No, 8], [3, 2, 4, [Yes, Yes, No], No, 12]], 6, [1])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 3]], [2, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 2, [Yes, Yes, Yes], Yes, 6], [2, 2, 2, [Yes, Yes, No], No, 8], [3, 2, 4, [Yes, Yes, No], No, 12]], 6, [1])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0091","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59ab563042953d602c42c4ee7fe83bacd51fd9f46e4fcc753f3bfaf33bfe3240","family_id":"qf-app21-lp-finite-optimum","seed":202610216091,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,4]],\"direction\":\"min\",\"objective\":[-2,2],\"points\":[[0,0],[1,3],[2,0],[3,2]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 3], [2, 0], [3, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 4]], objective=[-2, 2], direction=min. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 4]], [-2, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 3, [Yes, Yes, Yes], Yes, 4], [2, 2, 0, [Yes, Yes, Yes], Yes, -4], [3, 3, 2, [Yes, Yes, No], No, -2]], -4, [2])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 4]], [-2, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 3, [Yes, Yes, Yes], Yes, 4], [2, 2, 0, [Yes, Yes, Yes], Yes, -4], [3, 3, 2, [Yes, Yes, No], No, -2]], -4, [2])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0092","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3813066ebf1f48117bb17860c7f8d26aef1628d8926d498b056dc15d51f6b35e","family_id":"qf-app21-lp-finite-optimum","seed":202610216092,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,5]],\"direction\":\"max\",\"objective\":[-1,2],\"points\":[[0,0],[1,0],[2,1],[4,2]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 0], [2, 1], [4, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 5]], objective=[-1, 2], direction=max. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 5]], [-1, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 0, [Yes, Yes, Yes], Yes, -1], [2, 2, 1, [Yes, Yes, Yes], Yes, 0], [3, 4, 2, [Yes, Yes, No], No, 0]], 0, [0, 2])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 5]], [-1, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 0, [Yes, Yes, Yes], Yes, -1], [2, 2, 1, [Yes, Yes, Yes], Yes, 0], [3, 4, 2, [Yes, Yes, No], No, 0]], 0, [0, 2])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0093","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b57be1eecfba1fd9d3fe06d9c1d88d9af95c04a198fd267e3710bac9bc19836a","family_id":"qf-app21-lp-finite-optimum","seed":202610216093,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,6]],\"direction\":\"min\",\"objective\":[0,2],\"points\":[[0,0],[1,1],[2,2],[5,2]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 1], [2, 2], [5, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 6]], objective=[0, 2], direction=min. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 6]], [0, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 1, [Yes, Yes, Yes], Yes, 2], [2, 2, 2, [Yes, Yes, Yes], Yes, 4], [3, 5, 2, [Yes, Yes, No], No, 4]], 0, [0])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 6]], [0, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 1, [Yes, Yes, Yes], Yes, 2], [2, 2, 2, [Yes, Yes, Yes], Yes, 4], [3, 5, 2, [Yes, Yes, No], No, 4]], 0, [0])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0094","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"030302a499730f6b01c7f7dc71ff0eb0afd116a35a8c51ef205b242e2607b5e3","family_id":"qf-app21-lp-finite-optimum","seed":202610216094,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,7]],\"direction\":\"max\",\"objective\":[1,2],\"points\":[[0,0],[0,2],[1,2],[2,3]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 2], [2, 3], [0, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 7]], objective=[1, 2], direction=max. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 7]], [1, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 0, 2, [Yes, Yes, Yes], Yes, 4], [2, 1, 2, [Yes, Yes, Yes], Yes, 5], [3, 2, 3, [Yes, Yes, Yes], Yes, 8]], 8, [3])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 7]], [1, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 0, 2, [Yes, Yes, Yes], Yes, 4], [2, 1, 2, [Yes, Yes, Yes], Yes, 5], [3, 2, 3, [Yes, Yes, Yes], Yes, 8]], 8, [3])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0095","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"936c1bdf39fd6a917af3e1d96846057bdd4ceaa91175dc177512480010e602b6","family_id":"qf-app21-lp-finite-optimum","seed":202610216095,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,8]],\"direction\":\"min\",\"objective\":[2,2],\"points\":[[0,0],[1,2],[1,3],[2,4]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 3], [2, 4], [1, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 8]], objective=[2, 2], direction=min. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 8]], [2, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 2, [Yes, Yes, Yes], Yes, 6], [2, 1, 3, [Yes, Yes, Yes], Yes, 8], [3, 2, 4, [Yes, Yes, Yes], Yes, 12]], 0, [0])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 8]], [2, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 2, [Yes, Yes, Yes], Yes, 6], [2, 1, 3, [Yes, Yes, Yes], Yes, 8], [3, 2, 4, [Yes, Yes, Yes], Yes, 12]], 0, [0])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0096","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c8908d5bc5437bdb6440bf08644467be4b2e0875ea3582611da03f97e1c5076","family_id":"qf-app21-lp-finite-optimum","seed":202610216096,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,9]],\"direction\":\"max\",\"objective\":[-2,2],\"points\":[[0,0],[1,0],[2,0],[2,2]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 0], [2, 0], [2, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 9]], objective=[-2, 2], direction=max. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 9]], [-2, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 0, [Yes, Yes, Yes], Yes, -2], [2, 2, 0, [Yes, Yes, Yes], Yes, -4], [3, 2, 2, [Yes, Yes, Yes], Yes, 0]], 0, [0, 3])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 9]], [-2, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 0, [Yes, Yes, Yes], Yes, -2], [2, 2, 0, [Yes, Yes, Yes], Yes, -4], [3, 2, 2, [Yes, Yes, Yes], Yes, 0]], 0, [0, 3])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0097","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e9bf103c27768c3d97387c26b3ed22065b55f4c57f4305da82c59a30e67db14","family_id":"qf-app21-lp-finite-optimum","seed":202610216097,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,3]],\"direction\":\"min\",\"objective\":[-1,2],\"points\":[[0,0],[1,1],[2,1],[3,2]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 1], [2, 1], [3, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 3]], objective=[-1, 2], direction=min. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 3]], [-1, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 1, [Yes, Yes, Yes], Yes, 1], [2, 2, 1, [Yes, Yes, Yes], Yes, 0], [3, 3, 2, [Yes, Yes, No], No, 1]], 0, [0, 2])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 3]], [-1, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 1, [Yes, Yes, Yes], Yes, 1], [2, 2, 1, [Yes, Yes, Yes], Yes, 0], [3, 3, 2, [Yes, Yes, No], No, 1]], 0, [0, 2])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0098","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a491175f16f8a00d8a24b1d341454c28e4b9a88e8f2c8da7c31352ddb145bb2","family_id":"qf-app21-lp-finite-optimum","seed":202610216098,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,4]],\"direction\":\"max\",\"objective\":[0,2],\"points\":[[0,0],[1,2],[2,2],[4,2]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 2], [2, 2], [4, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 4]], objective=[0, 2], direction=max. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 4]], [0, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 2, [Yes, Yes, Yes], Yes, 4], [2, 2, 2, [Yes, Yes, Yes], Yes, 4], [3, 4, 2, [Yes, Yes, No], No, 4]], 4, [1, 2])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 4]], [0, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 2, [Yes, Yes, Yes], Yes, 4], [2, 2, 2, [Yes, Yes, Yes], Yes, 4], [3, 4, 2, [Yes, Yes, No], No, 4]], 4, [1, 2])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0099","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2f610e875552b54f01705c811015d1de30a72ed370355686e246a0dc91a6270","family_id":"qf-app21-lp-finite-optimum","seed":202610216099,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,5]],\"direction\":\"min\",\"objective\":[1,2],\"points\":[[0,0],[1,3],[2,3],[5,2]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 3], [2, 3], [5, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 5]], objective=[1, 2], direction=min. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 5]], [1, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 3, [Yes, Yes, Yes], Yes, 7], [2, 2, 3, [Yes, Yes, Yes], Yes, 8], [3, 5, 2, [Yes, Yes, No], No, 9]], 0, [0])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 5]], [1, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 3, [Yes, Yes, Yes], Yes, 7], [2, 2, 3, [Yes, Yes, Yes], Yes, 8], [3, 5, 2, [Yes, Yes, No], No, 9]], 0, [0])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r06-linear-programming-0100","batch_id":"batch-20261013-advance-20261021-r06-linear-programming-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"701afb1109c3480dfcc25caebcd98f1470aed7573d949083b3279687bb522743","family_id":"qf-app21-lp-finite-optimum","seed":202610216100,"semantic_template":"qf-app21-lp-finite-optimum/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,6]],\"direction\":\"max\",\"objective\":[2,2],\"points\":[[0,0],[0,2],[1,0],[2,4]]}"},"fields":{"title":"All Tied Optimizers among the Explicit Feasible Candidates","stem":"Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 0], [2, 4], [0, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 6]], objective=[2, 2], direction=max. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).","answer":{"kind":"expression","value":"([[-1, 0, 0], [0, -1, 0], [1, 1, 6]], [2, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 0, 2, [Yes, Yes, Yes], Yes, 4], [2, 1, 0, [Yes, Yes, Yes], Yes, 2], [3, 2, 4, [Yes, Yes, Yes], Yes, 12]], 12, [3])","explanation":"Compare only feasible rows, preserve all ties and restrict the optimality claim to the supplied finite grid. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 6]], [2, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 0, 2, [Yes, Yes, Yes], Yes, 4], [2, 1, 0, [Yes, Yes, Yes], Yes, 2], [3, 2, 4, [Yes, Yes, Yes], Yes, 12]], 12, [3])."},"skills":["All Tied Optimizers among the Explicit Feasible Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}