{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r06-similarity-0001","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff078abd5698fa342cb7845f6e6ed85f4f6d39f32f2c40f4fc761518b9a766cf","family_id":"qf-similarity-specified-scale","seed":202610076001,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"1/1\",\"1/1\",\"1/1\"],\"1/1\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(6, -4), B3=(4, -6), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=1/1. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"Yes","explanation":"The squared sides are A=[9, 5, 8] and B=[9, 5, 8]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0002","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90f48c7d7cdecc6bab1a8e8761891effab4143ce35e0a4b37e05cc09414e0ec3","family_id":"qf-similarity-specified-scale","seed":202610076002,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"4/1\",\"4/1\",\"4/1\"],\"3/1\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(9, -4), B3=(5, -8), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=3/1. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"No","explanation":"The squared sides are A=[9, 5, 8] and B=[36, 20, 32]. Check B²q²=A²p² on each corresponding side, obtaining No. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0003","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"522999be6df9559461047566549c81804dff8aa7a14ef18b660de77513c99e77","family_id":"qf-similarity-specified-scale","seed":202610076003,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"9/1\",\"9/1\",\"9/1\"],\"3/1\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(12, -4), B3=(6, -10), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=3/1. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"Yes","explanation":"The squared sides are A=[9, 5, 8] and B=[81, 45, 72]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0004","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c8c0e17809679d83d1cf8d967bbb403ad555595ad24abfb3e13a11207cd638b","family_id":"qf-similarity-specified-scale","seed":202610076004,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"49/4\",\"49/4\",\"49/4\"],\"7/2\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(6, 0), A3=(2, 4) and B1=(3, -4), B2=(24, -4), B3=(10, -18), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=7/2. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"Yes","explanation":"The squared sides are A=[36, 20, 32] and B=[441, 245, 392]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0005","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34fe0a082377b835d378116ad1f0461b2b1cd5cbdf401d1f9848edea6e498f77","family_id":"qf-similarity-specified-scale","seed":202610076005,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"25/1\",\"25/1\",\"25/1\"],\"5/1\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(18, -4), B3=(8, -14), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=5/1. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"Yes","explanation":"The squared sides are A=[9, 5, 8] and B=[225, 125, 200]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0006","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08979a73e2641803aec3c0fb9d2e02b05b59350ab33e2a74488754852259d9d0","family_id":"qf-similarity-specified-scale","seed":202610076006,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"36/1\",\"36/1\",\"36/1\"],\"7/1\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(21, -4), B3=(9, -16), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=7/1. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"No","explanation":"The squared sides are A=[9, 5, 8] and B=[324, 180, 288]. Check B²q²=A²p² on each corresponding side, obtaining No. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0007","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e7aa9ab7db6791634958b2f7051d1bad3532846f21179fe81071d1143bdf91f","family_id":"qf-similarity-specified-scale","seed":202610076007,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"49/1\",\"52/1\",\"365/8\"],\"7/1\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(24, -4), B3=(11, -18), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=7/1. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"No","explanation":"The squared sides are A=[9, 5, 8] and B=[441, 260, 365]. Check B²q²=A²p² on each corresponding side, obtaining No. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0008","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3d64cd1ae7dd19a45e5b8b7ecdd4a91047da3c731f562570eb360d5f998226c","family_id":"qf-similarity-specified-scale","seed":202610076008,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"64/1\",\"64/1\",\"64/1\"],\"9/1\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(27, -4), B3=(11, -20), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=9/1. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"No","explanation":"The squared sides are A=[9, 5, 8] and B=[576, 320, 512]. Check B²q²=A²p² on each corresponding side, obtaining No. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0009","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bbb4356e49717ed78cd08cf958b99b5ddab029fa3cbfa8a1f1a0a9ed3c654b17","family_id":"qf-similarity-specified-scale","seed":202610076009,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"289/4\",\"289/4\",\"289/4\"],\"17/2\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(6, 0), A3=(2, 4) and B1=(3, -4), B2=(54, -4), B3=(20, -38), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=17/2. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"Yes","explanation":"The squared sides are A=[36, 20, 32] and B=[2601, 1445, 2312]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0010","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a00e1ea9efe1a395f311f10124839a679f9484513e9db77eec47583bb0d17f94","family_id":"qf-similarity-specified-scale","seed":202610076010,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"100/1\",\"100/1\",\"100/1\"],\"11/1\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(33, -4), B3=(13, -24), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=11/1. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"No","explanation":"The squared sides are A=[9, 5, 8] and B=[900, 500, 800]. Check B²q²=A²p² on each corresponding side, obtaining No. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0011","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa0d04f7f11cea00a3d91d02f2b6f2c11db37dbd4cb6d2a07eebc9a50670bce5","family_id":"qf-similarity-specified-scale","seed":202610076011,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"121/1\",\"121/1\",\"121/1\"],\"11/1\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(36, -4), B3=(14, -26), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=11/1. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"Yes","explanation":"The squared sides are A=[9, 5, 8] and B=[1089, 605, 968]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0012","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa890732f6fa4df666f81750057fa149f07ba67e89f5f56b8b8d24b13c17e04e","family_id":"qf-similarity-specified-scale","seed":202610076012,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"144/1\",\"144/1\",\"144/1\"],\"13/1\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(39, -4), B3=(15, -28), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=13/1. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"No","explanation":"The squared sides are A=[9, 5, 8] and B=[1296, 720, 1152]. Check B²q²=A²p² on each corresponding side, obtaining No. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0013","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7315bbd46e0e63128511ecfddaa5e6afe062bc97cde6a36d3706ea57eb398d11","family_id":"qf-similarity-specified-scale","seed":202610076013,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"169/1\",\"169/1\",\"169/1\"],\"13/1\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(42, -4), B3=(16, -30), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=13/1. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"Yes","explanation":"The squared sides are A=[9, 5, 8] and B=[1521, 845, 1352]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0014","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc9b72bd088165cfa661c6d8527ca3da82d59c4da89f8dc2e04c47bb25fc73a5","family_id":"qf-similarity-specified-scale","seed":202610076014,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"729/4\",\"185/1\",\"5725/32\"],\"27/2\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(6, 0), A3=(2, 4) and B1=(3, -4), B2=(84, -4), B3=(31, -58), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=27/2. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"No","explanation":"The squared sides are A=[36, 20, 32] and B=[6561, 3700, 5725]. Check B²q²=A²p² on each corresponding side, obtaining No. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0015","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0cec1ba1325b864f3219b103afb003561449c86ff079cc3770c7d32f3ac9625","family_id":"qf-similarity-specified-scale","seed":202610076015,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"225/1\",\"225/1\",\"225/1\"],\"15/1\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(48, -4), B3=(18, -34), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=15/1. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"Yes","explanation":"The squared sides are A=[9, 5, 8] and B=[2025, 1125, 1800]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0016","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4de4a5b512c766a1e22b3c59249008c920fc06f19a266f1de29fa43bb04b895","family_id":"qf-similarity-specified-scale","seed":202610076016,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"256/1\",\"256/1\",\"256/1\"],\"17/1\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(51, -4), B3=(19, -36), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=17/1. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"No","explanation":"The squared sides are A=[9, 5, 8] and B=[2304, 1280, 2048]. Check B²q²=A²p² on each corresponding side, obtaining No. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0017","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3cea099d694b859110419193803308b142992c27f664f19af85e22629cff106","family_id":"qf-similarity-specified-scale","seed":202610076017,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"289/1\",\"289/1\",\"289/1\"],\"17/1\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(54, -4), B3=(20, -38), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=17/1. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"Yes","explanation":"The squared sides are A=[9, 5, 8] and B=[2601, 1445, 2312]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0018","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17f98726aac36ba43a4f7394aa9de9810d195990c5794f3ace5afc12b620598d","family_id":"qf-similarity-specified-scale","seed":202610076018,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"324/1\",\"324/1\",\"324/1\"],\"19/1\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(57, -4), B3=(21, -40), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=19/1. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"No","explanation":"The squared sides are A=[9, 5, 8] and B=[2916, 1620, 2592]. Check B²q²=A²p² on each corresponding side, obtaining No. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0019","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83d543fead98f5d83fdacb4beb695b5a9eb32d125304e6af9b82e5eb591830ad","family_id":"qf-similarity-specified-scale","seed":202610076019,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"1369/4\",\"1369/4\",\"1369/4\"],\"37/2\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(6, 0), A3=(2, 4) and B1=(3, -4), B2=(114, -4), B3=(40, -78), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=37/2. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"Yes","explanation":"The squared sides are A=[36, 20, 32] and B=[12321, 6845, 10952]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0020","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f1d4159949d4448d2defb095722a82b2dba6c56682d3888580c0de8c1aca9ac","family_id":"qf-similarity-specified-scale","seed":202610076020,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"400/1\",\"400/1\",\"400/1\"],\"21/1\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(63, -4), B3=(23, -44), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=21/1. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"No","explanation":"The squared sides are A=[9, 5, 8] and B=[3600, 2000, 3200]. Check B²q²=A²p² on each corresponding side, obtaining No. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0021","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f21855c99d98479d8ba03444d7104a3980a842447d6a74204be1b14fef87f4a","family_id":"qf-similarity-specified-scale","seed":202610076021,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"441/1\",\"2248/5\",\"3445/8\"],\"21/1\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(66, -4), B3=(25, -46), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=21/1. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"No","explanation":"The squared sides are A=[9, 5, 8] and B=[3969, 2248, 3445]. Check B²q²=A²p² on each corresponding side, obtaining No. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0022","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e7c082e707ab16434f11ae985a7a3cfeefd746a752df081e7b9367d19bb0169","family_id":"qf-similarity-specified-scale","seed":202610076022,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"484/1\",\"484/1\",\"484/1\"],\"23/1\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(69, -4), B3=(25, -48), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=23/1. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"No","explanation":"The squared sides are A=[9, 5, 8] and B=[4356, 2420, 3872]. Check B²q²=A²p² on each corresponding side, obtaining No. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0023","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a18958646b90d89500411a0782340e209ff19687a0663c3af7c28194316ea89","family_id":"qf-similarity-specified-scale","seed":202610076023,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"529/1\",\"529/1\",\"529/1\"],\"23/1\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(72, -4), B3=(26, -50), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=23/1. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"Yes","explanation":"The squared sides are A=[9, 5, 8] and B=[4761, 2645, 4232]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0024","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"660db279d4c6982c39f99c54545a421b8e3e42fe67fa1e877f4cf0e51bc63922","family_id":"qf-similarity-specified-scale","seed":202610076024,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"2209/4\",\"2209/4\",\"2209/4\"],\"47/2\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(6, 0), A3=(2, 4) and B1=(3, -4), B2=(144, -4), B3=(50, -98), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=47/2. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"Yes","explanation":"The squared sides are A=[36, 20, 32] and B=[19881, 11045, 17672]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0025","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00a84e775a9eb50c0daac4ce9e92c618393fc11a2ef2fc77fcb729587fa3c9cf","family_id":"qf-similarity-specified-scale","seed":202610076025,"semantic_template":"qf-similarity-specified-scale/{\"inputs\":[[\"625/1\",\"625/1\",\"625/1\"],\"25/1\"],\"operation\":\"specified_similarity\"}"},"fields":{"title":"Labeled Similarity at a Specified Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(78, -4), B3=(28, -54), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=25/1. Does this scale fit every corresponding side?","answer":{"kind":"short_text","value":"Yes","explanation":"The squared sides are A=[9, 5, 8] and B=[5625, 3125, 5000]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."},"skills":["Labeled Similarity at a Specified Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0026","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a563f22c98aa38e5715f9a42ed9d51867ea9be5dc5074795a4b2bfc99ea1fee7","family_id":"qf-similarity-positive-ratio","seed":202610076026,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"-12/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=-12/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"No","explanation":"The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0027","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"253459c6efcc98cdbfc6a9d020556af4630405b19fecf75cbad89d3f83eb7dfa","family_id":"qf-similarity-positive-ratio","seed":202610076027,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"-11/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=-11/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"No","explanation":"The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0028","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19cfa2cff5ee42361bd219b815f87945c692e5906851ac05a297edd61dbe8809","family_id":"qf-similarity-positive-ratio","seed":202610076028,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"-10/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=-10/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"No","explanation":"The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0029","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2cb90d36dd2614c053a49e60a078eb570a8bc28ed668bb5cbb88442cc531f72","family_id":"qf-similarity-positive-ratio","seed":202610076029,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"-9/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=-9/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"No","explanation":"The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0030","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34307c6460de390e648b6573d63d5bb8a51bd31879e808e22a02f1dc19f7c192","family_id":"qf-similarity-positive-ratio","seed":202610076030,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"-8/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=-8/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"No","explanation":"The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0031","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e03baef2cd5166edf83c7ba4e84a74591e77afcd51eef96eef635f6ca74259f","family_id":"qf-similarity-positive-ratio","seed":202610076031,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"-7/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=-7/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"No","explanation":"The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0032","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"760cfc315652e9bc955cff327c97332df5a4ffada5d365d08f11a9ff93a3f0e1","family_id":"qf-similarity-positive-ratio","seed":202610076032,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"-6/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=-6/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"No","explanation":"The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0033","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89bd3948fe5d2135bc6b8cc11d6b749a287e1a59436f028633d4e1d9a7e11f03","family_id":"qf-similarity-positive-ratio","seed":202610076033,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"-5/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=-5/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"No","explanation":"The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0034","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bb43e62c02154cd1dd24b23ce2f4f40274bcb74d4ed3ada69abfb2cd3fca560","family_id":"qf-similarity-positive-ratio","seed":202610076034,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"-4/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=-4/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"No","explanation":"The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0035","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78f835203e2091c3bc14d9cdc489f8bf4e38859384c6fb8e3ff1e7236ed5b414","family_id":"qf-similarity-positive-ratio","seed":202610076035,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"-3/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=-3/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"No","explanation":"The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0036","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b38e1a0286e10396e29c284dfee52abd7ca7368dd43381d917d8e48625d19c8b","family_id":"qf-similarity-positive-ratio","seed":202610076036,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"-2/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=-2/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"No","explanation":"The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0037","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0b296f2d626db9478a009b74efa25df919195c8d61c1f69c4a54194f59aa60f","family_id":"qf-similarity-positive-ratio","seed":202610076037,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"-1/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=-1/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"No","explanation":"The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0038","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"254330879a745c7847f0f209a961e40631c5dcc22a8c74dbf8aace1cf7e26f7f","family_id":"qf-similarity-positive-ratio","seed":202610076038,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"0/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=0/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"No","explanation":"The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0039","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b17c350557fd1432b87dcb33610b1037428208c46a6dfe400ecdc4d5e9af1f2","family_id":"qf-similarity-positive-ratio","seed":202610076039,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"1/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=1/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"Yes","explanation":"The denominator is valid, and the numerator must also be positive, giving Yes. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0040","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f28bfd15060ffbfd6e916d817753517ac0a9d0094eeff6f1af232f2b173bc13d","family_id":"qf-similarity-positive-ratio","seed":202610076040,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"2/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=2/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"Yes","explanation":"The denominator is valid, and the numerator must also be positive, giving Yes. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0041","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72557fe25a9a1da238c8e010d5f8ae41097c139d1898de369e525361b4e69ddb","family_id":"qf-similarity-positive-ratio","seed":202610076041,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"3/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=3/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"Yes","explanation":"The denominator is valid, and the numerator must also be positive, giving Yes. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0042","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"208e5844fd2251509ae8b3a82a9dbcab3d18f6396dec37c3f8b28d26b6baba9e","family_id":"qf-similarity-positive-ratio","seed":202610076042,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"4/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=4/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"Yes","explanation":"The denominator is valid, and the numerator must also be positive, giving Yes. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0043","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"213ef1045e9313e0116755de5ae055444bd0b3929300edadc44268d408b142d9","family_id":"qf-similarity-positive-ratio","seed":202610076043,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"5/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=5/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"Yes","explanation":"The denominator is valid, and the numerator must also be positive, giving Yes. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0044","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"748b0a9b6b1dd6ba3e12f7e835201ea8b8ed00686e0b617094287ba921bb1b09","family_id":"qf-similarity-positive-ratio","seed":202610076044,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"6/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=6/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"Yes","explanation":"The denominator is valid, and the numerator must also be positive, giving Yes. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0045","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e20890d05e253afe9af97485b4c0c1f534d86c08dfc6f69857dd710f7ce1b845","family_id":"qf-similarity-positive-ratio","seed":202610076045,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"7/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=7/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"Yes","explanation":"The denominator is valid, and the numerator must also be positive, giving Yes. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0046","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d09b04ff3f4882018c80f9fb62295354acce3b2b8eacfa6f50942b4bd8314b36","family_id":"qf-similarity-positive-ratio","seed":202610076046,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"8/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=8/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"Yes","explanation":"The denominator is valid, and the numerator must also be positive, giving Yes. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0047","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cee8676a979e8efc40f15fb506f49b162069f1d786b4456829708adc5b9044a","family_id":"qf-similarity-positive-ratio","seed":202610076047,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"9/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=9/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"Yes","explanation":"The denominator is valid, and the numerator must also be positive, giving Yes. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0048","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70814af9371c770177ff60c990ba7935b05b67adaa6c9978ad72e0580af9fc76","family_id":"qf-similarity-positive-ratio","seed":202610076048,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"10/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=10/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"Yes","explanation":"The denominator is valid, and the numerator must also be positive, giving Yes. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0049","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46ddef59b4f66439d0cc58225891cb670781efd727c70ca4de9229972cc864e2","family_id":"qf-similarity-positive-ratio","seed":202610076049,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"11/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=11/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"Yes","explanation":"The denominator is valid, and the numerator must also be positive, giving Yes. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0050","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc381022ff9bd16d5d836f83678e15b41f2a57512c23745052153985c67a3c33","family_id":"qf-similarity-positive-ratio","seed":202610076050,"semantic_template":"qf-similarity-positive-ratio/{\"inputs\":[\"12/1\"],\"operation\":\"positive_scale_ratio\"}"},"fields":{"title":"Positive Length-Scale Eligibility","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=12/1 an eligible positive length scale for nondegenerate similarity?","answer":{"kind":"short_text","value":"Yes","explanation":"The denominator is valid, and the numerator must also be positive, giving Yes. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients."},"skills":["Positive Length-Scale Eligibility"],"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-20261006-advance-20261007-r06-similarity-0051","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93da47a685dff82d2bc6eb7f77a58de966e884dfc30433b43d32b8328625e39f","family_id":"qf-similarity-scaled-squared-side","seed":202610076051,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"2\",\"1/1\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 2>0. At corresponding positive length scale k=1/1, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"2/1","explanation":"Multiply the given positive squared length by k² to obtain 2/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0052","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1eee335f7ac81e0d589d33642dc854a10e8c59d385536f88a18475fed75413f","family_id":"qf-similarity-scaled-squared-side","seed":202610076052,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"5\",\"1/1\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 5>0. At corresponding positive length scale k=2/2, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"5/1","explanation":"Multiply the given positive squared length by k² to obtain 5/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0053","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55709a9dc8975d9034369b7cbafdab8f7cf89a1a5dc56c3f710c2fd9202f1c35","family_id":"qf-similarity-scaled-squared-side","seed":202610076053,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"10\",\"1/1\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 10>0. At corresponding positive length scale k=3/3, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"10/1","explanation":"Multiply the given positive squared length by k² to obtain 10/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0054","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b7d11941c59b516afa41ded4f1ff4025136f68c6501762be0d2d6db9b4368a3","family_id":"qf-similarity-scaled-squared-side","seed":202610076054,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"17\",\"4/1\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 17>0. At corresponding positive length scale k=4/1, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"272/1","explanation":"Multiply the given positive squared length by k² to obtain 272/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0055","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f827894c2426c219e5fe31f3e4356ded752b4141300f54482e7460e981cfd6fd","family_id":"qf-similarity-scaled-squared-side","seed":202610076055,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"26\",\"5/2\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 26>0. At corresponding positive length scale k=5/2, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"325/2","explanation":"Multiply the given positive squared length by k² to obtain 325/2. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0056","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64a2e5ff4744a18b5788afc4d76507d4db1b689902436e2761e67c318e3b2eb9","family_id":"qf-similarity-scaled-squared-side","seed":202610076056,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"37\",\"1/3\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 37>0. At corresponding positive length scale k=1/3, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"37/9","explanation":"Multiply the given positive squared length by k² to obtain 37/9. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0057","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16e69b7f0636234906f653d2d9eb0da28b1098b3beb07863f90bdad49be4593e","family_id":"qf-similarity-scaled-squared-side","seed":202610076057,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"50\",\"2/1\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 50>0. At corresponding positive length scale k=2/1, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"200/1","explanation":"Multiply the given positive squared length by k² to obtain 200/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0058","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46067ef7c50f6fcbe23b9efbeab8e1e437d7fe541d94d0b6077e34a27d2caa8e","family_id":"qf-similarity-scaled-squared-side","seed":202610076058,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"65\",\"3/2\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 65>0. At corresponding positive length scale k=3/2, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"585/4","explanation":"Multiply the given positive squared length by k² to obtain 585/4. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0059","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6d8733c3e2069431b63f945223c6ce4410eff51d461cfb0d93464ebc24cba0d","family_id":"qf-similarity-scaled-squared-side","seed":202610076059,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"82\",\"4/3\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 82>0. At corresponding positive length scale k=4/3, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"1312/9","explanation":"Multiply the given positive squared length by k² to obtain 1312/9. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0060","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8bf4471f9afa05c605d9bcc08aa96148ac6afe6c0958c0e6b0979a94d330824","family_id":"qf-similarity-scaled-squared-side","seed":202610076060,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"101\",\"5/1\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 101>0. At corresponding positive length scale k=5/1, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"2525/1","explanation":"Multiply the given positive squared length by k² to obtain 2525/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0061","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fba455a2df02c361456426e38aa88aec7742c6cc47c4d2658dc31b92f576a97e","family_id":"qf-similarity-scaled-squared-side","seed":202610076061,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"122\",\"1/2\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 122>0. At corresponding positive length scale k=1/2, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"61/2","explanation":"Multiply the given positive squared length by k² to obtain 61/2. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0062","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a36f17ca812c2992e42722b29fa6411e970d4ee9d30c63c128504a8e185fa59f","family_id":"qf-similarity-scaled-squared-side","seed":202610076062,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"145\",\"2/3\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 145>0. At corresponding positive length scale k=2/3, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"580/9","explanation":"Multiply the given positive squared length by k² to obtain 580/9. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0063","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eebfb1517e53426f8171135151e6c20aa30a8ae90d76f677f716fd29d6ba15cb","family_id":"qf-similarity-scaled-squared-side","seed":202610076063,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"170\",\"3/1\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 170>0. At corresponding positive length scale k=3/1, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"1530/1","explanation":"Multiply the given positive squared length by k² to obtain 1530/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0064","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba30a1b9acf1f225e43add0757553a59caf327a5f9cd5274688ddacb812fabf9","family_id":"qf-similarity-scaled-squared-side","seed":202610076064,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"197\",\"2/1\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 197>0. At corresponding positive length scale k=4/2, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"788/1","explanation":"Multiply the given positive squared length by k² to obtain 788/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0065","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75ff805aff6745ab7f7b56114d5073173201355861f155008983c442332db11d","family_id":"qf-similarity-scaled-squared-side","seed":202610076065,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"226\",\"5/3\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 226>0. At corresponding positive length scale k=5/3, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"5650/9","explanation":"Multiply the given positive squared length by k² to obtain 5650/9. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0066","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eed2b1d121acfb312440e0c20b10e3209c06ec030305fbfa34488aa3fcff410e","family_id":"qf-similarity-scaled-squared-side","seed":202610076066,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"257\",\"1/1\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 257>0. At corresponding positive length scale k=1/1, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"257/1","explanation":"Multiply the given positive squared length by k² to obtain 257/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0067","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c92f6e7d1479b9f514b1cb21556e04c086d0b0155c57a2ab8ee38ec02bbe7624","family_id":"qf-similarity-scaled-squared-side","seed":202610076067,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"290\",\"1/1\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 290>0. At corresponding positive length scale k=2/2, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"290/1","explanation":"Multiply the given positive squared length by k² to obtain 290/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0068","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a98c8702eefba67a3d036235fd22e4e3225dbcccd2f30f89f2b4d565e9817c9","family_id":"qf-similarity-scaled-squared-side","seed":202610076068,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"325\",\"1/1\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 325>0. At corresponding positive length scale k=3/3, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"325/1","explanation":"Multiply the given positive squared length by k² to obtain 325/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0069","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d4a25d67ee271a37e1ea192430ce9310b843829fe8a558eb77e6b95d53026ff","family_id":"qf-similarity-scaled-squared-side","seed":202610076069,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"362\",\"4/1\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 362>0. At corresponding positive length scale k=4/1, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"5792/1","explanation":"Multiply the given positive squared length by k² to obtain 5792/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0070","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e4dd59b3a1e9ed24aff247d12be197f3073ec725da842d197ded6f3d53dc59c","family_id":"qf-similarity-scaled-squared-side","seed":202610076070,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"401\",\"5/2\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 401>0. At corresponding positive length scale k=5/2, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"10025/4","explanation":"Multiply the given positive squared length by k² to obtain 10025/4. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0071","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e9e2edea9722cfdf4f517b37420ede42fc175132586b4f8a494b0b7400cf06b","family_id":"qf-similarity-scaled-squared-side","seed":202610076071,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"442\",\"1/3\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 442>0. At corresponding positive length scale k=1/3, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"442/9","explanation":"Multiply the given positive squared length by k² to obtain 442/9. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0072","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1aad183af0bdb86d56bc879e779119f1e7973d65cab32d4817cde071f0ff4262","family_id":"qf-similarity-scaled-squared-side","seed":202610076072,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"485\",\"2/1\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 485>0. At corresponding positive length scale k=2/1, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"1940/1","explanation":"Multiply the given positive squared length by k² to obtain 1940/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0073","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"382f6ebef3440827464bfb22b1d811784a2e2c60e7c1d5d956e6ae23834e9f38","family_id":"qf-similarity-scaled-squared-side","seed":202610076073,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"530\",\"3/2\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 530>0. At corresponding positive length scale k=3/2, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"2385/2","explanation":"Multiply the given positive squared length by k² to obtain 2385/2. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0074","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90493ca30ecc1ffe2ab33933f405fd9bc2dcf2dff3597404909d3874c425c1a6","family_id":"qf-similarity-scaled-squared-side","seed":202610076074,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"577\",\"4/3\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 577>0. At corresponding positive length scale k=4/3, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"9232/9","explanation":"Multiply the given positive squared length by k² to obtain 9232/9. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0075","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f04051ffa1abed72a02e82d6660dbf946b4fcb9ca4624e718ea24f3258c19fc","family_id":"qf-similarity-scaled-squared-side","seed":202610076075,"semantic_template":"qf-similarity-scaled-squared-side/{\"inputs\":[\"626\",\"5/1\"],\"operation\":\"scaled_squared_side\"}"},"fields":{"title":"Squared Side Length under a Positive Scale","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 626>0. At corresponding positive length scale k=5/1, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"15650/1","explanation":"Multiply the given positive squared length by k² to obtain 15650/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity."},"skills":["Squared Side Length under a Positive Scale"],"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-20261006-advance-20261007-r06-similarity-0076","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7f85e34aa78b674cdb8c271e8fa2f28dc3f676d8a5648db156d28de9b28afc1","family_id":"qf-similarity-scaled-area","seed":202610076076,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"1/2\",\"1/1\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 1/2>0 squared units. At corresponding positive length scale k=1/1, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"1/2","explanation":"Multiply the given positive area by k² to obtain 1/2 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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-20261006-advance-20261007-r06-similarity-0077","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a3fdc4ffb2771dfb921621db947cb2877e2d5d80e99caadbac6d9e2c49451bf","family_id":"qf-similarity-scaled-area","seed":202610076077,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"3/2\",\"1/1\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 3/2>0 squared units. At corresponding positive length scale k=2/2, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"3/2","explanation":"Multiply the given positive area by k² to obtain 3/2 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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-20261006-advance-20261007-r06-similarity-0078","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a05e2d780b3d0a39d14d4a1a896805f06307ec620d09b196df9cc9fea78b01fd","family_id":"qf-similarity-scaled-area","seed":202610076078,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"5/2\",\"1/1\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 5/2>0 squared units. At corresponding positive length scale k=3/3, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"5/2","explanation":"Multiply the given positive area by k² to obtain 5/2 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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-20261006-advance-20261007-r06-similarity-0079","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"caa71498faaa16451da07e2116604d3ae002fed72fc5a399cfacb3e27e411272","family_id":"qf-similarity-scaled-area","seed":202610076079,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"7/2\",\"4/1\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 7/2>0 squared units. At corresponding positive length scale k=4/1, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"56/1","explanation":"Multiply the given positive area by k² to obtain 56/1 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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-20261006-advance-20261007-r06-similarity-0080","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1409c27aa22902df3d66cac9609462a4f601ad60c5f8cbddc064526259f86f69","family_id":"qf-similarity-scaled-area","seed":202610076080,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"9/2\",\"1/2\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 9/2>0 squared units. At corresponding positive length scale k=1/2, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"9/8","explanation":"Multiply the given positive area by k² to obtain 9/8 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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-20261006-advance-20261007-r06-similarity-0081","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ef2c291393d4ac8dfe1a6f8f07ff49fc4d1fe7ed58e3d474e0b957c98ce5187","family_id":"qf-similarity-scaled-area","seed":202610076081,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"11/2\",\"2/3\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 11/2>0 squared units. At corresponding positive length scale k=2/3, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"22/9","explanation":"Multiply the given positive area by k² to obtain 22/9 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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-20261006-advance-20261007-r06-similarity-0082","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43f971af056ff71576803b02d0748151d71c746b06c943866653fb42dfd50b36","family_id":"qf-similarity-scaled-area","seed":202610076082,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"13/2\",\"3/1\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 13/2>0 squared units. At corresponding positive length scale k=3/1, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"117/2","explanation":"Multiply the given positive area by k² to obtain 117/2 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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-20261006-advance-20261007-r06-similarity-0083","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa60d7e5876b92a79c118d04a78d1fc70966ef5715afef3f85e79909de29458f","family_id":"qf-similarity-scaled-area","seed":202610076083,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"15/2\",\"2/1\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 15/2>0 squared units. At corresponding positive length scale k=4/2, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"30/1","explanation":"Multiply the given positive area by k² to obtain 30/1 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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-20261006-advance-20261007-r06-similarity-0084","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"faa9e6beb8db296eff9742395cd5177996a304686c38a61a4f54e78a85775c4a","family_id":"qf-similarity-scaled-area","seed":202610076084,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"17/2\",\"1/3\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 17/2>0 squared units. At corresponding positive length scale k=1/3, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"17/18","explanation":"Multiply the given positive area by k² to obtain 17/18 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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-20261006-advance-20261007-r06-similarity-0085","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2d1fbc095ed562208b3be3807a7688b6d31e26f81fab218f15d1bbe01dcd3e9","family_id":"qf-similarity-scaled-area","seed":202610076085,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"19/2\",\"2/1\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 19/2>0 squared units. At corresponding positive length scale k=2/1, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"38/1","explanation":"Multiply the given positive area by k² to obtain 38/1 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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-20261006-advance-20261007-r06-similarity-0086","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7a45549de7ff4c04a2e3504c3d3f155763c5696837181d94ad11faae72a78eb","family_id":"qf-similarity-scaled-area","seed":202610076086,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"21/2\",\"3/2\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 21/2>0 squared units. At corresponding positive length scale k=3/2, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"189/8","explanation":"Multiply the given positive area by k² to obtain 189/8 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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-20261006-advance-20261007-r06-similarity-0087","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2874e655aed7ab039c546e94b8298fcde7182811d4bbaab1498812af9499c667","family_id":"qf-similarity-scaled-area","seed":202610076087,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"23/2\",\"4/3\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 23/2>0 squared units. At corresponding positive length scale k=4/3, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"184/9","explanation":"Multiply the given positive area by k² to obtain 184/9 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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-20261006-advance-20261007-r06-similarity-0088","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"228070c39818f8e0f4cd9216122b1c53f32182caa172c8fd7750337a0309be8a","family_id":"qf-similarity-scaled-area","seed":202610076088,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"25/2\",\"1/1\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 25/2>0 squared units. At corresponding positive length scale k=1/1, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"25/2","explanation":"Multiply the given positive area by k² to obtain 25/2 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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-20261006-advance-20261007-r06-similarity-0089","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db51d4e6b574a8da7bd66764d04308ff052e9d76ca409f2c56f4b5ba54d74e81","family_id":"qf-similarity-scaled-area","seed":202610076089,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"27/2\",\"1/1\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 27/2>0 squared units. At corresponding positive length scale k=2/2, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"27/2","explanation":"Multiply the given positive area by k² to obtain 27/2 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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-20261006-advance-20261007-r06-similarity-0090","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b656dc13b206abe44031b5759eb59863873d7eef83420227ed615c85ffed87d8","family_id":"qf-similarity-scaled-area","seed":202610076090,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"29/2\",\"1/1\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 29/2>0 squared units. At corresponding positive length scale k=3/3, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"29/2","explanation":"Multiply the given positive area by k² to obtain 29/2 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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-20261006-advance-20261007-r06-similarity-0091","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3eec9bfa9cf11d3465a4ef2a364b3a3987b5f5d581eeabd1406c06d7b89e83e1","family_id":"qf-similarity-scaled-area","seed":202610076091,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"31/2\",\"4/1\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 31/2>0 squared units. At corresponding positive length scale k=4/1, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"248/1","explanation":"Multiply the given positive area by k² to obtain 248/1 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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-20261006-advance-20261007-r06-similarity-0092","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82728ccaa65ab8738984e0717ab9d92583616013df3352892055e2e8ac689b8f","family_id":"qf-similarity-scaled-area","seed":202610076092,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"33/2\",\"1/2\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 33/2>0 squared units. At corresponding positive length scale k=1/2, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"33/8","explanation":"Multiply the given positive area by k² to obtain 33/8 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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-20261006-advance-20261007-r06-similarity-0093","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33d6232e191d6b3896bdcc21982a478184203c34cb3d34de6e2f84ff0fda25bb","family_id":"qf-similarity-scaled-area","seed":202610076093,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"35/2\",\"2/3\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 35/2>0 squared units. At corresponding positive length scale k=2/3, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"70/9","explanation":"Multiply the given positive area by k² to obtain 70/9 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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-20261006-advance-20261007-r06-similarity-0094","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06b9b18a478d2bc7bcd4ab7dcf110a1d9112ab37847ac67df6f10386d618dbf9","family_id":"qf-similarity-scaled-area","seed":202610076094,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"37/2\",\"3/1\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 37/2>0 squared units. At corresponding positive length scale k=3/1, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"333/2","explanation":"Multiply the given positive area by k² to obtain 333/2 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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-20261006-advance-20261007-r06-similarity-0095","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bf75464e7f98f6293b0710c5b1163b700d8d6e84bc7ea6d24cbca1bad2bbc2d","family_id":"qf-similarity-scaled-area","seed":202610076095,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"39/2\",\"2/1\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 39/2>0 squared units. At corresponding positive length scale k=4/2, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"78/1","explanation":"Multiply the given positive area by k² to obtain 78/1 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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-20261006-advance-20261007-r06-similarity-0096","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"499b7a23f4e75074042d569f180815804a0718f855ba9b481aa53909645a58cc","family_id":"qf-similarity-scaled-area","seed":202610076096,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"41/2\",\"1/3\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 41/2>0 squared units. At corresponding positive length scale k=1/3, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"41/18","explanation":"Multiply the given positive area by k² to obtain 41/18 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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-20261006-advance-20261007-r06-similarity-0097","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1e1e9030b125de1f9abeb7ce3f025446a3ec9c69281886ed3420b1649d6d099","family_id":"qf-similarity-scaled-area","seed":202610076097,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"43/2\",\"2/1\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 43/2>0 squared units. At corresponding positive length scale k=2/1, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"86/1","explanation":"Multiply the given positive area by k² to obtain 86/1 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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-20261006-advance-20261007-r06-similarity-0098","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45e9088a1315f2699f85844e6e48335caee676c35ec3d53dd5e2cc04c796dc03","family_id":"qf-similarity-scaled-area","seed":202610076098,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"45/2\",\"3/2\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 45/2>0 squared units. At corresponding positive length scale k=3/2, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"405/8","explanation":"Multiply the given positive area by k² to obtain 405/8 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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-20261006-advance-20261007-r06-similarity-0099","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c3a72d195dc451987fa313296e38d1e7944a99ccdf9252c8ff013f16de8d566","family_id":"qf-similarity-scaled-area","seed":202610076099,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"47/2\",\"4/3\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 47/2>0 squared units. At corresponding positive length scale k=4/3, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"376/9","explanation":"Multiply the given positive area by k² to obtain 376/9 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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-20261006-advance-20261007-r06-similarity-0100","batch_id":"batch-20261006-advance-20261007-r06-similarity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e17202b98e8eff32046fe079a0a8336b6320aebb55cb6bcf6457b138fdb5ea89","family_id":"qf-similarity-scaled-area","seed":202610076100,"semantic_template":"qf-similarity-scaled-area/{\"inputs\":[\"49/2\",\"1/1\"],\"operation\":\"scaled_area\"}"},"fields":{"title":"Conditional Scaling of a Given Positive Area","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 49/2>0 squared units. At corresponding positive length scale k=1/1, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"49/2","explanation":"Multiply the given positive area by k² to obtain 49/2 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory."},"skills":["Conditional Scaling of a Given Positive Area"],"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."}}