{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r05-circle-0001","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2ae5c2b118b5a7566f7f2831af32491f9f15c08e5369524428e1fdc5f670b91","family_id":"qf-circle-membership","seed":202610065001,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"0\"],\"0\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-12, 5) and radius r=0≥0. For P=(-12, 5), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare squared distance 0 with r²=0; equality is required, giving Yes. Strictly inside points are not boundary members. At zero radius the boundary degenerates to the singleton center, with no strictly inside points; all axial representations coincide."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0002","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f0d2f4410ae77ef8f03e7bcf63ea659cc9cda5b5d6f04752a4b826ee7dcce23","family_id":"qf-circle-membership","seed":202610065002,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"1\"],\"0\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-11, 4) and radius r=0≥0. For P=(-10, 4), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"No","explanation":"Compare squared distance 1 with r²=0; equality is required, giving No. Strictly inside points are not boundary members. At zero radius the boundary degenerates to the singleton center, with no strictly inside points; all axial representations coincide."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0003","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"298e8432e350bc4b0bcbde1bafdf48bac94f10e08fe9f4a29d7bbd6ec369622b","family_id":"qf-circle-membership","seed":202610065003,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"1\",\"2\"],\"1\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-10, 3) and radius r=1≥0. For P=(-8, 4), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"No","explanation":"Compare squared distance 5 with r²=1; equality is required, giving No. Strictly inside points are not boundary members."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0004","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48990e1ec7bcf9f59cae7dd6729041cf14308855c6995a7ff9556d47f3d02e42","family_id":"qf-circle-membership","seed":202610065004,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"2\"],\"2\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-9, 2) and radius r=2≥0. For P=(-7, 2), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare squared distance 4 with r²=4; equality is required, giving Yes. Strictly inside points are not boundary members."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0005","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4ca3f75768b0bac7cd74a7ce3da86a8ec9c929a9d19a6018998e3d1deb6aa16","family_id":"qf-circle-membership","seed":202610065005,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"2\"],\"3\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-8, 1) and radius r=3≥0. For P=(-8, 3), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"No","explanation":"Compare squared distance 4 with r²=9; equality is required, giving No. Strictly inside points are not boundary members."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0006","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9231b124aae5814be6c58d9aa02b521a4a7727ea7aae599953c1a5188ee1a7d7","family_id":"qf-circle-membership","seed":202610065006,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"1\",\"5\"],\"4\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-7, 0) and radius r=4≥0. For P=(-2, 1), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"No","explanation":"Compare squared distance 26 with r²=16; equality is required, giving No. Strictly inside points are not boundary members."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0007","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e0b539b20011de82f59338822b2ea3451518c2abcf23f62ccab219f94c7f727","family_id":"qf-circle-membership","seed":202610065007,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"5\"],\"5\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-6, -1) and radius r=5≥0. For P=(-1, -1), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare squared distance 25 with r²=25; equality is required, giving Yes. Strictly inside points are not boundary members."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0008","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b16e07b8175b4e9b4e35dbad0cd489eef777b3bace4a41e3044aa29a35e706a","family_id":"qf-circle-membership","seed":202610065008,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"5\"],\"6\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-5, -2) and radius r=6≥0. For P=(-5, 3), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"No","explanation":"Compare squared distance 25 with r²=36; equality is required, giving No. Strictly inside points are not boundary members."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0009","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a3892ed840f158eca0280305310d797720dc53d5d1fd5e870ec723ac076d1ac","family_id":"qf-circle-membership","seed":202610065009,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"1\",\"8\"],\"7\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-4, -3) and radius r=7≥0. For P=(4, -2), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"No","explanation":"Compare squared distance 65 with r²=49; equality is required, giving No. Strictly inside points are not boundary members."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0010","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbcb2f5e1d5dfe8ae98119cad5655cb689bbed434f00bf6f9ebff7737e06fee6","family_id":"qf-circle-membership","seed":202610065010,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"8\"],\"8\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-3, -4) and radius r=8≥0. For P=(5, -4), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare squared distance 64 with r²=64; equality is required, giving Yes. Strictly inside points are not boundary members."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0011","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea139e32620bf4eab33bd303db129629667d403c0d53931ed2a49ed6b493f079","family_id":"qf-circle-membership","seed":202610065011,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"8\"],\"9\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-2, -5) and radius r=9≥0. For P=(-2, 3), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"No","explanation":"Compare squared distance 64 with r²=81; equality is required, giving No. Strictly inside points are not boundary members."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0012","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fde149a9ab0d521234536695f2578911a3e16760ac144876495fb18d3cb01e83","family_id":"qf-circle-membership","seed":202610065012,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"1\",\"11\"],\"10\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-1, -6) and radius r=10≥0. For P=(10, -5), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"No","explanation":"Compare squared distance 122 with r²=100; equality is required, giving No. Strictly inside points are not boundary members."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0013","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54ceefdbce0fbf6535e40dac4f197db6cc1678607b68d0a0e6d4fb6c31478412","family_id":"qf-circle-membership","seed":202610065013,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"11\"],\"11\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(0, -7) and radius r=11≥0. For P=(11, -7), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare squared distance 121 with r²=121; equality is required, giving Yes. Strictly inside points are not boundary members."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0014","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ae41908deed02e9037d7437e4040786c689816b980d43c70f48e1f1f40c0f78","family_id":"qf-circle-membership","seed":202610065014,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"11\"],\"12\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(1, -8) and radius r=12≥0. For P=(1, 3), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"No","explanation":"Compare squared distance 121 with r²=144; equality is required, giving No. Strictly inside points are not boundary members."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0015","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ceb9ed98d202b8dbc3c4c96bcec7497ce4e032c7be83d3a0257f9bc90d28521b","family_id":"qf-circle-membership","seed":202610065015,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"1\",\"14\"],\"13\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(2, -9) and radius r=13≥0. For P=(16, -8), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"No","explanation":"Compare squared distance 197 with r²=169; equality is required, giving No. Strictly inside points are not boundary members."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0016","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"625a8f38e844119686257f2d7b7ec0e207dd7d55ddc8d5f3e3057a025d0bc83e","family_id":"qf-circle-membership","seed":202610065016,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"14\"],\"14\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(3, -10) and radius r=14≥0. For P=(17, -10), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare squared distance 196 with r²=196; equality is required, giving Yes. Strictly inside points are not boundary members."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0017","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4f0441848a4a99508e261425ba43064fcf38a98e6407b52b6e82adb01309869","family_id":"qf-circle-membership","seed":202610065017,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"14\"],\"15\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(4, -11) and radius r=15≥0. For P=(4, 3), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"No","explanation":"Compare squared distance 196 with r²=225; equality is required, giving No. Strictly inside points are not boundary members."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0018","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1805e92a2e5f832272ff9c1876486708f1f2f6faef38a4b29b5e3637ba7b9598","family_id":"qf-circle-membership","seed":202610065018,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"1\",\"17\"],\"16\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(5, -12) and radius r=16≥0. For P=(22, -11), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"No","explanation":"Compare squared distance 290 with r²=256; equality is required, giving No. Strictly inside points are not boundary members."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0019","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0475fea652a4246e80fac139c423d22ab733b90a8b0214a42125756a0ddad6e3","family_id":"qf-circle-membership","seed":202610065019,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"17\"],\"17\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(6, -13) and radius r=17≥0. For P=(23, -13), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare squared distance 289 with r²=289; equality is required, giving Yes. Strictly inside points are not boundary members."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0020","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ed4e78c9f9dcd8ed23a11d9d00c748975af6e677b77788aa82f9071553d5101","family_id":"qf-circle-membership","seed":202610065020,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"17\"],\"18\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(7, -14) and radius r=18≥0. For P=(7, 3), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"No","explanation":"Compare squared distance 289 with r²=324; equality is required, giving No. Strictly inside points are not boundary members."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0021","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4028eeb085f812e98b4e70fad3376d7560a8d106266e8a96a23cdf9d31843bc6","family_id":"qf-circle-membership","seed":202610065021,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"1\",\"20\"],\"19\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(8, -15) and radius r=19≥0. For P=(28, -14), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"No","explanation":"Compare squared distance 401 with r²=361; equality is required, giving No. Strictly inside points are not boundary members."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0022","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffc7e631f96c30f45ec4a5e02190aa6a0f0fc8749cbcb8d0442ad8e48e515372","family_id":"qf-circle-membership","seed":202610065022,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"20\"],\"20\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(9, -16) and radius r=20≥0. For P=(29, -16), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare squared distance 400 with r²=400; equality is required, giving Yes. Strictly inside points are not boundary members."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0023","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5160905ccb1f41c937701b0d6b2807e6f6ca1150beb57386d7a86d74a8671cce","family_id":"qf-circle-membership","seed":202610065023,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"20\"],\"21\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(10, -17) and radius r=21≥0. For P=(10, 3), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"No","explanation":"Compare squared distance 400 with r²=441; equality is required, giving No. Strictly inside points are not boundary members."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0024","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d25ca8c28cce88ba945b5c8107dd8d762c7b5c31542968a3f97281ceb5857238","family_id":"qf-circle-membership","seed":202610065024,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"1\",\"23\"],\"22\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(11, -18) and radius r=22≥0. For P=(34, -17), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"No","explanation":"Compare squared distance 530 with r²=484; equality is required, giving No. Strictly inside points are not boundary members."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0025","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"330178faaf1253448f2bbd2b614dbc7e962d1968c46b7c1fa9053f180ff8031c","family_id":"qf-circle-membership","seed":202610065025,"semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"23\"],\"23\"],\"operation\":\"circle_membership\"}"},"fields":{"title":"Membership in a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(12, -19) and radius r=23≥0. For P=(35, -19), is P a member of the circle boundary, not merely of the filled disk?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare squared distance 529 with r²=529; equality is required, giving Yes. Strictly inside points are not boundary members."},"skills":["Membership in a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0026","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fd06050039f753d7e37f0a2c3b290f7644e446d8135eeaa73043d54be09b435","family_id":"qf-circle-position","seed":202610065026,"semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"0\"],\"0\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-12, 5) and radius r=0≥0. Classify P=(-12, 5) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"on boundary","explanation":"Compare squared distance 0 with r²=0: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary. At zero radius the boundary degenerates to the singleton center, with no strictly inside points; all axial representations coincide."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0027","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6854beb77890f9d6624f552694d7fce20a43d2d645a519abdf6305ba4138847","family_id":"qf-circle-position","seed":202610065027,"semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"1\"],\"0\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-11, 4) and radius r=0≥0. Classify P=(-10, 4) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"outside","explanation":"Compare squared distance 1 with r²=0: less means inside, equal means on boundary, and greater means outside. Here the result is outside. At zero radius the boundary degenerates to the singleton center, with no strictly inside points; all axial representations coincide."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0028","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8aac17038d3097d0b9750b563be5b07c9e84d8ded568cf0f2d392bfecefecb8","family_id":"qf-circle-position","seed":202610065028,"semantic_template":"qf-circle-position/{\"inputs\":[[\"1\",\"2\"],\"1\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-10, 3) and radius r=1≥0. Classify P=(-8, 4) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"outside","explanation":"Compare squared distance 5 with r²=1: less means inside, equal means on boundary, and greater means outside. Here the result is outside."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0029","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f53f240140a777a036044fee70b2fa1572cd28d8cf22fef25badfa15dc2bbb62","family_id":"qf-circle-position","seed":202610065029,"semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"2\"],\"2\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-9, 2) and radius r=2≥0. Classify P=(-7, 2) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"on boundary","explanation":"Compare squared distance 4 with r²=4: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0030","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28061d04d9d0c5dc6ab01fb8b2520ba8003b2707872f461ab9bb11d5575b6ce0","family_id":"qf-circle-position","seed":202610065030,"semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"2\"],\"3\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-8, 1) and radius r=3≥0. Classify P=(-8, 3) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"inside","explanation":"Compare squared distance 4 with r²=9: less means inside, equal means on boundary, and greater means outside. Here the result is inside."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0031","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04c6532f4df21d0096e7f05820652dfcf1fdcea92d6a60aac3c27bf52c291ccb","family_id":"qf-circle-position","seed":202610065031,"semantic_template":"qf-circle-position/{\"inputs\":[[\"1\",\"5\"],\"4\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-7, 0) and radius r=4≥0. Classify P=(-2, 1) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"outside","explanation":"Compare squared distance 26 with r²=16: less means inside, equal means on boundary, and greater means outside. Here the result is outside."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0032","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"505255659b4769acba1b4d56a423c6446fc8c61deb9b9e34c909c84f916498d2","family_id":"qf-circle-position","seed":202610065032,"semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"5\"],\"5\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-6, -1) and radius r=5≥0. Classify P=(-1, -1) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"on boundary","explanation":"Compare squared distance 25 with r²=25: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0033","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa5dfcab116b81363bbb14d8624c9a29eb0eda24ba71be6c85a2afa2dc55f931","family_id":"qf-circle-position","seed":202610065033,"semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"5\"],\"6\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-5, -2) and radius r=6≥0. Classify P=(-5, 3) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"inside","explanation":"Compare squared distance 25 with r²=36: less means inside, equal means on boundary, and greater means outside. Here the result is inside."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0034","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d3ad2e6d7cc8add91b11fadae5ecc018d6e1b1acd5a9594e6794f7d38ba59e2","family_id":"qf-circle-position","seed":202610065034,"semantic_template":"qf-circle-position/{\"inputs\":[[\"1\",\"8\"],\"7\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-4, -3) and radius r=7≥0. Classify P=(4, -2) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"outside","explanation":"Compare squared distance 65 with r²=49: less means inside, equal means on boundary, and greater means outside. Here the result is outside."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0035","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bdb73397e4d5c9a4df05750b762c011aa10260efacc6931f4c2c10006fef2e2","family_id":"qf-circle-position","seed":202610065035,"semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"8\"],\"8\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-3, -4) and radius r=8≥0. Classify P=(5, -4) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"on boundary","explanation":"Compare squared distance 64 with r²=64: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0036","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99a0822cb19ebcc00011ef25f7fd431b4c712966cc12960ae39b3187bcefe58e","family_id":"qf-circle-position","seed":202610065036,"semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"8\"],\"9\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-2, -5) and radius r=9≥0. Classify P=(-2, 3) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"inside","explanation":"Compare squared distance 64 with r²=81: less means inside, equal means on boundary, and greater means outside. Here the result is inside."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0037","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e98bbadc3fb98f930c8dc4931bb7e9b6be4cabb27d792c0ec68e03622294bf1","family_id":"qf-circle-position","seed":202610065037,"semantic_template":"qf-circle-position/{\"inputs\":[[\"1\",\"11\"],\"10\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-1, -6) and radius r=10≥0. Classify P=(10, -5) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"outside","explanation":"Compare squared distance 122 with r²=100: less means inside, equal means on boundary, and greater means outside. Here the result is outside."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0038","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"526132590c65d2056af8b72baca5793d03398c21113520c95fd0cea0d6326593","family_id":"qf-circle-position","seed":202610065038,"semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"11\"],\"11\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(0, -7) and radius r=11≥0. Classify P=(11, -7) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"on boundary","explanation":"Compare squared distance 121 with r²=121: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0039","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"132cd70da49a4f448810f41b27c0ea8f6f3ba123a45adbfad9c9e639cf50f4b4","family_id":"qf-circle-position","seed":202610065039,"semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"11\"],\"12\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(1, -8) and radius r=12≥0. Classify P=(1, 3) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"inside","explanation":"Compare squared distance 121 with r²=144: less means inside, equal means on boundary, and greater means outside. Here the result is inside."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0040","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50e21dd37b42739ad788ff0553f689683a82da9517145151a2ce5e75d433125c","family_id":"qf-circle-position","seed":202610065040,"semantic_template":"qf-circle-position/{\"inputs\":[[\"1\",\"14\"],\"13\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(2, -9) and radius r=13≥0. Classify P=(16, -8) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"outside","explanation":"Compare squared distance 197 with r²=169: less means inside, equal means on boundary, and greater means outside. Here the result is outside."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0041","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60dd550d583b0ba9e36b27308b3665d117e3bf70440cf71ed86887b2e735406d","family_id":"qf-circle-position","seed":202610065041,"semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"14\"],\"14\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(3, -10) and radius r=14≥0. Classify P=(17, -10) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"on boundary","explanation":"Compare squared distance 196 with r²=196: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0042","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bf4d4ce9baf4e1724c7ca61458a4977f1d7039278566892e527cd060f0d5cfa","family_id":"qf-circle-position","seed":202610065042,"semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"14\"],\"15\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(4, -11) and radius r=15≥0. Classify P=(4, 3) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"inside","explanation":"Compare squared distance 196 with r²=225: less means inside, equal means on boundary, and greater means outside. Here the result is inside."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0043","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57300657208d3c7d0a6d75bb4d3bc0f5dbdeaca5340a0c11790cd7819d5e42f7","family_id":"qf-circle-position","seed":202610065043,"semantic_template":"qf-circle-position/{\"inputs\":[[\"1\",\"17\"],\"16\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(5, -12) and radius r=16≥0. Classify P=(22, -11) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"outside","explanation":"Compare squared distance 290 with r²=256: less means inside, equal means on boundary, and greater means outside. Here the result is outside."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0044","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c88c99fa59d6ab96869b3a5f06d9389428eb5d97adefd570e19b3e238e42bb6f","family_id":"qf-circle-position","seed":202610065044,"semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"17\"],\"17\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(6, -13) and radius r=17≥0. Classify P=(23, -13) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"on boundary","explanation":"Compare squared distance 289 with r²=289: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0045","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"789b9db11cecb8867a8e093ebb1090a71a3d564b3052b977dcaa02cfd6d9b879","family_id":"qf-circle-position","seed":202610065045,"semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"17\"],\"18\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(7, -14) and radius r=18≥0. Classify P=(7, 3) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"inside","explanation":"Compare squared distance 289 with r²=324: less means inside, equal means on boundary, and greater means outside. Here the result is inside."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0046","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d30634b30785c6d0c73ee6d4c31398b3b6328e6503907cdb3de9ed8015daf534","family_id":"qf-circle-position","seed":202610065046,"semantic_template":"qf-circle-position/{\"inputs\":[[\"1\",\"20\"],\"19\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(8, -15) and radius r=19≥0. Classify P=(28, -14) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"outside","explanation":"Compare squared distance 401 with r²=361: less means inside, equal means on boundary, and greater means outside. Here the result is outside."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0047","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bb21287f02209bb73da7e705d7ec49148c5017678016f21534de0ba7b4234f8","family_id":"qf-circle-position","seed":202610065047,"semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"20\"],\"20\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(9, -16) and radius r=20≥0. Classify P=(29, -16) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"on boundary","explanation":"Compare squared distance 400 with r²=400: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0048","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad2c2115122521a5b898588830d1955662f466b7e1d0ec344582edf88a8a740c","family_id":"qf-circle-position","seed":202610065048,"semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"20\"],\"21\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(10, -17) and radius r=21≥0. Classify P=(10, 3) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"inside","explanation":"Compare squared distance 400 with r²=441: less means inside, equal means on boundary, and greater means outside. Here the result is inside."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0049","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a3af56f7fa6091da3aa9c28da57c62734b604da419d9af67391979e736843f3","family_id":"qf-circle-position","seed":202610065049,"semantic_template":"qf-circle-position/{\"inputs\":[[\"1\",\"23\"],\"22\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(11, -18) and radius r=22≥0. Classify P=(34, -17) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"outside","explanation":"Compare squared distance 530 with r²=484: less means inside, equal means on boundary, and greater means outside. Here the result is outside."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0050","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e44f86b65f06811b6a9227e02f0331b7d494b92efff4f5cc9ef5df41ee3f781f","family_id":"qf-circle-position","seed":202610065050,"semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"23\"],\"23\"],\"operation\":\"circle_position\"}"},"fields":{"title":"Three Positions Relative to a Circle Boundary","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(12, -19) and radius r=23≥0. Classify P=(35, -19) as inside, on boundary, or outside.","answer":{"kind":"short_text","value":"on boundary","explanation":"Compare squared distance 529 with r²=529: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary."},"skills":["Three Positions Relative to a Circle Boundary"],"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-20261005-advance-20261006-r05-circle-0051","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c98c47dbab1d05536510186036ecd224e20863c126b7c3e60f3fa3be83a42ab","family_id":"qf-circle-diameter","seed":202610065051,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"0\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=0≥0. Find its diameter length.","answer":{"kind":"short_text","value":"0","explanation":"The diameter length is 2r=0 in the same length unit, including zero at radius zero. At zero radius the boundary degenerates to the singleton center, with no strictly inside points; all axial representations coincide."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0052","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62e9bf7fb92d842e8f8d4b520cf54c37dfbb35b272a59ad64880cfce7292e2e3","family_id":"qf-circle-diameter","seed":202610065052,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"1\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=1≥0. Find its diameter length.","answer":{"kind":"short_text","value":"2","explanation":"The diameter length is 2r=2 in the same length unit, including zero at radius zero."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0053","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b2c6a79486aa6af0e45a12cdad229c39b9edb7ebfad69800a00b0fd5b74e7f2","family_id":"qf-circle-diameter","seed":202610065053,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"2\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=2≥0. Find its diameter length.","answer":{"kind":"short_text","value":"4","explanation":"The diameter length is 2r=4 in the same length unit, including zero at radius zero."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0054","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d596ddb41c45fc138403cf8e9a694b9ee56fb0ee109e630f8b0d90cf6841120a","family_id":"qf-circle-diameter","seed":202610065054,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"3\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=3≥0. Find its diameter length.","answer":{"kind":"short_text","value":"6","explanation":"The diameter length is 2r=6 in the same length unit, including zero at radius zero."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0055","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6514ec4e88235b08fce9927c53cc9eefa5733f0beab09069a8f8513da26a0130","family_id":"qf-circle-diameter","seed":202610065055,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"4\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=4≥0. Find its diameter length.","answer":{"kind":"short_text","value":"8","explanation":"The diameter length is 2r=8 in the same length unit, including zero at radius zero."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0056","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20d565a0c5b7005b8e6ebef5b1cb1a5611ab558e63cb83a4b0717c52cf190c23","family_id":"qf-circle-diameter","seed":202610065056,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"5\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=5≥0. Find its diameter length.","answer":{"kind":"short_text","value":"10","explanation":"The diameter length is 2r=10 in the same length unit, including zero at radius zero."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0057","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"755bb09dbe925f5b415e01bf933998c1164d27d3e9832beaa9491262f8e4b920","family_id":"qf-circle-diameter","seed":202610065057,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"6\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=6≥0. Find its diameter length.","answer":{"kind":"short_text","value":"12","explanation":"The diameter length is 2r=12 in the same length unit, including zero at radius zero."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0058","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"706180bc9b0eb05d7158bf0e0d2be4ad632420dcb1347963eb5ad13ddac39a2f","family_id":"qf-circle-diameter","seed":202610065058,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"7\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=7≥0. Find its diameter length.","answer":{"kind":"short_text","value":"14","explanation":"The diameter length is 2r=14 in the same length unit, including zero at radius zero."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0059","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e097162c30a6ae34f2c903bce17929b13a5ecc6f6bd0e85a52bfaa933ad0f523","family_id":"qf-circle-diameter","seed":202610065059,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"8\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=8≥0. Find its diameter length.","answer":{"kind":"short_text","value":"16","explanation":"The diameter length is 2r=16 in the same length unit, including zero at radius zero."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0060","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b9d51718fe35c3fbf9b4d7e3d65c4c35d1e045facf595b70bc141d955bbdf2f","family_id":"qf-circle-diameter","seed":202610065060,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"9\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=9≥0. Find its diameter length.","answer":{"kind":"short_text","value":"18","explanation":"The diameter length is 2r=18 in the same length unit, including zero at radius zero."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0061","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48fe63fd8741c799b896477756b235d607920007d4dfb447ef9a39925998741e","family_id":"qf-circle-diameter","seed":202610065061,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"10\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=10≥0. Find its diameter length.","answer":{"kind":"short_text","value":"20","explanation":"The diameter length is 2r=20 in the same length unit, including zero at radius zero."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0062","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5f2bc089844971f1a00012570806d6733f28468f583639a793bfa42773a29da","family_id":"qf-circle-diameter","seed":202610065062,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"11\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=11≥0. Find its diameter length.","answer":{"kind":"short_text","value":"22","explanation":"The diameter length is 2r=22 in the same length unit, including zero at radius zero."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0063","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b4c45fd9169013e72dbb8d6a74bcc5bca423bad71c91ccf4c0b70fc8589c03a","family_id":"qf-circle-diameter","seed":202610065063,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"12\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=12≥0. Find its diameter length.","answer":{"kind":"short_text","value":"24","explanation":"The diameter length is 2r=24 in the same length unit, including zero at radius zero."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0064","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7930710e6603425023054af868e4062c807760fa653722ad92458bf06dd0662c","family_id":"qf-circle-diameter","seed":202610065064,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"13\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=13≥0. Find its diameter length.","answer":{"kind":"short_text","value":"26","explanation":"The diameter length is 2r=26 in the same length unit, including zero at radius zero."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0065","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79e13109d14aff7e0aecc820950b70a950d640ea2824cb405a9118159cca8586","family_id":"qf-circle-diameter","seed":202610065065,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"14\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=14≥0. Find its diameter length.","answer":{"kind":"short_text","value":"28","explanation":"The diameter length is 2r=28 in the same length unit, including zero at radius zero."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0066","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23c8e1d88179e5011992d1a25a7cbe5e50e93d1672fceaf1a2912f3ddc59f6f0","family_id":"qf-circle-diameter","seed":202610065066,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"15\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=15≥0. Find its diameter length.","answer":{"kind":"short_text","value":"30","explanation":"The diameter length is 2r=30 in the same length unit, including zero at radius zero."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0067","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"199490537f46d554ce5dbe3077919144fbea863c5f594dd4375fd0e32f309c22","family_id":"qf-circle-diameter","seed":202610065067,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"16\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=16≥0. Find its diameter length.","answer":{"kind":"short_text","value":"32","explanation":"The diameter length is 2r=32 in the same length unit, including zero at radius zero."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0068","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf9de287183981ac66191a906add555b4587256b783b76c5fdea1459fa6d16b8","family_id":"qf-circle-diameter","seed":202610065068,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"17\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=17≥0. Find its diameter length.","answer":{"kind":"short_text","value":"34","explanation":"The diameter length is 2r=34 in the same length unit, including zero at radius zero."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0069","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"935369edeceb913516089c9b94eafdf5005952872c2c7e7cd2b8086f0dd2cf07","family_id":"qf-circle-diameter","seed":202610065069,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"18\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=18≥0. Find its diameter length.","answer":{"kind":"short_text","value":"36","explanation":"The diameter length is 2r=36 in the same length unit, including zero at radius zero."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0070","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e9447fa0244bd442eabd210353fa4cb95febbb0bf4373a00f3cd0b45b3764d1","family_id":"qf-circle-diameter","seed":202610065070,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"19\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=19≥0. Find its diameter length.","answer":{"kind":"short_text","value":"38","explanation":"The diameter length is 2r=38 in the same length unit, including zero at radius zero."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0071","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86966202a6f844c50e8ba146e47652aa6faba83926f69476fb9dd35d0d4263e9","family_id":"qf-circle-diameter","seed":202610065071,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"20\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=20≥0. Find its diameter length.","answer":{"kind":"short_text","value":"40","explanation":"The diameter length is 2r=40 in the same length unit, including zero at radius zero."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0072","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fc9b9f41b5afe26bb674b0a7f771260338c079f586f82db98442f7878b699b5","family_id":"qf-circle-diameter","seed":202610065072,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"21\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=21≥0. Find its diameter length.","answer":{"kind":"short_text","value":"42","explanation":"The diameter length is 2r=42 in the same length unit, including zero at radius zero."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0073","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fab88a91578e65fe024580d170d14339f5f5bf8fa66901ba0424d04d527bd60d","family_id":"qf-circle-diameter","seed":202610065073,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"22\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=22≥0. Find its diameter length.","answer":{"kind":"short_text","value":"44","explanation":"The diameter length is 2r=44 in the same length unit, including zero at radius zero."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0074","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"899ceea114d9d69e6af446a26a8f7b0477881670d2c7ec07af9484a350c45969","family_id":"qf-circle-diameter","seed":202610065074,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"23\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=23≥0. Find its diameter length.","answer":{"kind":"short_text","value":"46","explanation":"The diameter length is 2r=46 in the same length unit, including zero at radius zero."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0075","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b213b91c69082577be2e9e384993aa112f3d4f0aef281c5a9a9d1186f57d40f","family_id":"qf-circle-diameter","seed":202610065075,"semantic_template":"qf-circle-diameter/{\"inputs\":[\"24\"],\"operation\":\"circle_diameter\"}"},"fields":{"title":"Diameter Length for a Nonnegative Radius","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=24≥0. Find its diameter length.","answer":{"kind":"short_text","value":"48","explanation":"The diameter length is 2r=48 in the same length unit, including zero at radius zero."},"skills":["Diameter Length for a Nonnegative Radius"],"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-20261005-advance-20261006-r05-circle-0076","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c64082006294d1441f5377f6e9f9f053352913debb8e9b13ab8ac40694dae9d1","family_id":"qf-circle-axial-point","seed":202610065076,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"0\",\"0\"],\"zero_radius\"],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(0, 0) and radius r=0≥0. Change only its y-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(0, 0)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (0, 0), exactly on the boundary rather than inferred from a sketch. At zero radius the boundary degenerates to the singleton center, with no strictly inside points; all axial representations coincide."},"skills":["Specified Axial Point on a Circle"],"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-20261005-advance-20261006-r05-circle-0077","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76d198ff2e490ed979cd56455c0ed116ac088b20780525290dd1d031fd4d2c36","family_id":"qf-circle-axial-point","seed":202610065077,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"-11\",\"2\"],\"1\",\"x\",1],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-11, 2) and radius r=1≥0. Change only its x-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(-10, 2)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-10, 2), exactly on the boundary rather than inferred from a sketch."},"skills":["Specified Axial Point on a Circle"],"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-20261005-advance-20261006-r05-circle-0078","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1049325c0a3f273271c8f740c67c0c5fb4a9c8fd76c6faf74186bf790ac7d83","family_id":"qf-circle-axial-point","seed":202610065078,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"-10\",\"1\"],\"2\",\"y\",1],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-10, 1) and radius r=2≥0. Change only its y-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(-10, 3)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-10, 3), exactly on the boundary rather than inferred from a sketch."},"skills":["Specified Axial Point on a Circle"],"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-20261005-advance-20261006-r05-circle-0079","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbbe9b41c07ed3cbd4dec8a6e72cd6dccc33790fe8711ede867781e52224df6b","family_id":"qf-circle-axial-point","seed":202610065079,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"-9\",\"0\"],\"3\",\"x\",-1],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-9, 0) and radius r=3≥0. Change only its x-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(-12, 0)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-12, 0), exactly on the boundary rather than inferred from a sketch."},"skills":["Specified Axial Point on a Circle"],"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-20261005-advance-20261006-r05-circle-0080","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29febf931ef02bbf8abbdc14177998b09cc795bc36fd37cdce100ca61128e902","family_id":"qf-circle-axial-point","seed":202610065080,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"-8\",\"-1\"],\"4\",\"y\",1],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-8, -1) and radius r=4≥0. Change only its y-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(-8, 3)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-8, 3), exactly on the boundary rather than inferred from a sketch."},"skills":["Specified Axial Point on a Circle"],"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-20261005-advance-20261006-r05-circle-0081","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1eeb8eb69d398554ab79e7c0e481eddc661ff727eb2a9f1bb03a024c09907c22","family_id":"qf-circle-axial-point","seed":202610065081,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"-7\",\"-2\"],\"5\",\"x\",1],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-7, -2) and radius r=5≥0. Change only its x-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(-2, -2)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-2, -2), exactly on the boundary rather than inferred from a sketch."},"skills":["Specified Axial Point on a Circle"],"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-20261005-advance-20261006-r05-circle-0082","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77dc795bd806e909b2734e3fc08c27f7df8a81e446941ba20784ef691a270b43","family_id":"qf-circle-axial-point","seed":202610065082,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"-6\",\"-3\"],\"6\",\"y\",-1],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-6, -3) and radius r=6≥0. Change only its y-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(-6, -9)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-6, -9), exactly on the boundary rather than inferred from a sketch."},"skills":["Specified Axial Point on a Circle"],"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-20261005-advance-20261006-r05-circle-0083","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71e1ffc2508fcad9eece7f4b2527d2af3ed030dff05087235d46d9551e260c5b","family_id":"qf-circle-axial-point","seed":202610065083,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"-5\",\"-4\"],\"7\",\"x\",1],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-5, -4) and radius r=7≥0. Change only its x-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(2, -4)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (2, -4), exactly on the boundary rather than inferred from a sketch."},"skills":["Specified Axial Point on a Circle"],"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-20261005-advance-20261006-r05-circle-0084","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb4bf0e132c7607f8c99a28003f5cb946169968e8301786ddd84434b239e53e8","family_id":"qf-circle-axial-point","seed":202610065084,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"-4\",\"-5\"],\"8\",\"y\",1],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-4, -5) and radius r=8≥0. Change only its y-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(-4, 3)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-4, 3), exactly on the boundary rather than inferred from a sketch."},"skills":["Specified Axial Point on a Circle"],"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-20261005-advance-20261006-r05-circle-0085","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aefc5696854ce468d877f671b43f71b9624f2d2ade22204e39fcb147d7bc1063","family_id":"qf-circle-axial-point","seed":202610065085,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"-3\",\"-6\"],\"9\",\"x\",-1],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-3, -6) and radius r=9≥0. Change only its x-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(-12, -6)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-12, -6), exactly on the boundary rather than inferred from a sketch."},"skills":["Specified Axial Point on a Circle"],"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-20261005-advance-20261006-r05-circle-0086","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"707d33f29e636243ee4ca05205a3dfe57ede548bccfa8a6c9ed35ce393c6b89a","family_id":"qf-circle-axial-point","seed":202610065086,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"-2\",\"-7\"],\"10\",\"y\",1],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-2, -7) and radius r=10≥0. Change only its y-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(-2, 3)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-2, 3), exactly on the boundary rather than inferred from a sketch."},"skills":["Specified Axial Point on a Circle"],"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-20261005-advance-20261006-r05-circle-0087","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"691eafe4eb3f961714394fe3242405b3fdfe81dfd7115167998b4733a5d42911","family_id":"qf-circle-axial-point","seed":202610065087,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"-1\",\"-8\"],\"11\",\"x\",1],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-1, -8) and radius r=11≥0. Change only its x-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(10, -8)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (10, -8), exactly on the boundary rather than inferred from a sketch."},"skills":["Specified Axial Point on a Circle"],"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-20261005-advance-20261006-r05-circle-0088","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64571dd286169f6b66698b3a930b422c565bb4f934e729276762639ec3fbc06c","family_id":"qf-circle-axial-point","seed":202610065088,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"0\",\"-9\"],\"12\",\"y\",-1],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(0, -9) and radius r=12≥0. Change only its y-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(0, -21)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (0, -21), exactly on the boundary rather than inferred from a sketch."},"skills":["Specified Axial Point on a Circle"],"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-20261005-advance-20261006-r05-circle-0089","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e6f4c26137a141c58645ad4d3b9c515c628f925b2fef4bbbfe87f678d5c016f","family_id":"qf-circle-axial-point","seed":202610065089,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"1\",\"-10\"],\"13\",\"x\",1],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(1, -10) and radius r=13≥0. Change only its x-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(14, -10)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (14, -10), exactly on the boundary rather than inferred from a sketch."},"skills":["Specified Axial Point on a Circle"],"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-20261005-advance-20261006-r05-circle-0090","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95f2a67e7e6ddc84a13d460df36fbb60b15a53c2fced6c6bc481e71fc19aad37","family_id":"qf-circle-axial-point","seed":202610065090,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"2\",\"-11\"],\"14\",\"y\",1],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(2, -11) and radius r=14≥0. Change only its y-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(2, 3)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (2, 3), exactly on the boundary rather than inferred from a sketch."},"skills":["Specified Axial Point on a Circle"],"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-20261005-advance-20261006-r05-circle-0091","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41e1cca14df33a079c056b6015a0f7ab46cc9e68a6f58bce148831415642d75f","family_id":"qf-circle-axial-point","seed":202610065091,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"3\",\"-12\"],\"15\",\"x\",-1],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(3, -12) and radius r=15≥0. Change only its x-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(-12, -12)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-12, -12), exactly on the boundary rather than inferred from a sketch."},"skills":["Specified Axial Point on a Circle"],"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-20261005-advance-20261006-r05-circle-0092","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9ab039343890f018b32b5f2c1b1102bb2ac97efa3ae6db7bdf158211210c2c9","family_id":"qf-circle-axial-point","seed":202610065092,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"4\",\"-13\"],\"16\",\"y\",1],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(4, -13) and radius r=16≥0. Change only its y-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(4, 3)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (4, 3), exactly on the boundary rather than inferred from a sketch."},"skills":["Specified Axial Point on a Circle"],"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-20261005-advance-20261006-r05-circle-0093","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f7526748275e251e8d74d09b653e72cb0995a0c9e53ddc37d56ca5bb9f597ea","family_id":"qf-circle-axial-point","seed":202610065093,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"5\",\"-14\"],\"17\",\"x\",1],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(5, -14) and radius r=17≥0. Change only its x-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(22, -14)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (22, -14), exactly on the boundary rather than inferred from a sketch."},"skills":["Specified Axial Point on a Circle"],"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-20261005-advance-20261006-r05-circle-0094","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de6e6e75f54beba1994b477ac8c42e2fcda9463babbbbe136a39c99291225f00","family_id":"qf-circle-axial-point","seed":202610065094,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"6\",\"-15\"],\"18\",\"y\",-1],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(6, -15) and radius r=18≥0. Change only its y-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(6, -33)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (6, -33), exactly on the boundary rather than inferred from a sketch."},"skills":["Specified Axial Point on a Circle"],"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-20261005-advance-20261006-r05-circle-0095","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3b91162792ce3e8a9f64be260136949593db4c9ca77990c32953c1ef36a6448","family_id":"qf-circle-axial-point","seed":202610065095,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"7\",\"-16\"],\"19\",\"x\",1],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(7, -16) and radius r=19≥0. Change only its x-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(26, -16)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (26, -16), exactly on the boundary rather than inferred from a sketch."},"skills":["Specified Axial Point on a Circle"],"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-20261005-advance-20261006-r05-circle-0096","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d101749cd9184e1ea369655dca48d4c6d9621b6b76d72d30eeefdb24dea7b972","family_id":"qf-circle-axial-point","seed":202610065096,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"8\",\"-17\"],\"20\",\"y\",1],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(8, -17) and radius r=20≥0. Change only its y-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(8, 3)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (8, 3), exactly on the boundary rather than inferred from a sketch."},"skills":["Specified Axial Point on a Circle"],"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-20261005-advance-20261006-r05-circle-0097","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06167e9c9d77a2e1b038e6a2871e22d145f1c07601ce66eb4bbfad66d7432f97","family_id":"qf-circle-axial-point","seed":202610065097,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"9\",\"-18\"],\"21\",\"x\",-1],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(9, -18) and radius r=21≥0. Change only its x-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(-12, -18)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-12, -18), exactly on the boundary rather than inferred from a sketch."},"skills":["Specified Axial Point on a Circle"],"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-20261005-advance-20261006-r05-circle-0098","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b06b1b7ac48872e32bf3f2928c12837a8cdec6a4db4dfc22fab1da5997aad90","family_id":"qf-circle-axial-point","seed":202610065098,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"10\",\"-19\"],\"22\",\"y\",1],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(10, -19) and radius r=22≥0. Change only its y-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(10, 3)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (10, 3), exactly on the boundary rather than inferred from a sketch."},"skills":["Specified Axial Point on a Circle"],"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-20261005-advance-20261006-r05-circle-0099","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24e87d320a3e46430d213d3290b96c9b4600916dd2315659964a69b822906943","family_id":"qf-circle-axial-point","seed":202610065099,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"11\",\"-20\"],\"23\",\"x\",1],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(11, -20) and radius r=23≥0. Change only its x-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(34, -20)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (34, -20), exactly on the boundary rather than inferred from a sketch."},"skills":["Specified Axial Point on a Circle"],"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-20261005-advance-20261006-r05-circle-0100","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"228d49145f288bfab556c885ea85a4f4ea5f3ac83e1778e94fcdc5b32f3b8de3","family_id":"qf-circle-axial-point","seed":202610065100,"semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"12\",\"-21\"],\"24\",\"y\",-1],\"operation\":\"circle_axial\"}"},"fields":{"title":"Specified Axial Point on a Circle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(12, -21) and radius r=24≥0. Change only its y-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.","answer":{"kind":"expression","value":"(12, -45)","explanation":"Add the signed radius to the specified coordinate and leave the other unchanged. The point is (12, -45), exactly on the boundary rather than inferred from a sketch."},"skills":["Specified Axial Point on a Circle"],"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."}}