{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r06-area-0001","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6b0fefd723f806f969f9e9c3b9c2ebcea0ebe55e7f33c44a6abf2fa0abfdd08","family_id":"qf-area-rectangle-dimensions","seed":202610066001,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"empty_rectangle\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,0)×[0,3) has width 0 and height 3, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"0","explanation":"Multiply width by height to obtain 0 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0002","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5695287bc024f45732e16b0bed0bc940992f3c9e8f6b1cf8d7e5c3db67f91526","family_id":"qf-area-rectangle-dimensions","seed":202610066002,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"1\",\"2\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,1)×[0,2) has width 1 and height 2, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"2","explanation":"Multiply width by height to obtain 2 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0003","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3144cfa885db17723dbc2f973d26e35863ba73fbeb74a85a23b3bcadbeac42b","family_id":"qf-area-rectangle-dimensions","seed":202610066003,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"2\",\"3\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,2)×[0,3) has width 2 and height 3, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"6","explanation":"Multiply width by height to obtain 6 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0004","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a082c1d6cc62b9205c2d6167336c10127ca9f28c4f72cc547b4d03c30db88200","family_id":"qf-area-rectangle-dimensions","seed":202610066004,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"3\",\"4\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,3)×[0,4) has width 3 and height 4, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"12","explanation":"Multiply width by height to obtain 12 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0005","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0b0b680ad09f06648410a877f9fd697f103a7e68e467ca331dde1c957961ba8","family_id":"qf-area-rectangle-dimensions","seed":202610066005,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"4\",\"5\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,4)×[0,5) has width 4 and height 5, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"20","explanation":"Multiply width by height to obtain 20 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0006","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49583d2e5e472a649333e8b9c03f46a4fd50e2a113c9fb2f7e8ef7d97f96c086","family_id":"qf-area-rectangle-dimensions","seed":202610066006,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"1\",\"5\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,5)×[0,1) has width 5 and height 1, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"5","explanation":"Multiply width by height to obtain 5 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0007","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ebdd37770877b6d1386063f5026e2db50f538692347bdd45b2d06655380d895","family_id":"qf-area-rectangle-dimensions","seed":202610066007,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"2\",\"6\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,6)×[0,2) has width 6 and height 2, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"12","explanation":"Multiply width by height to obtain 12 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0008","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d995f2fbeff6994f41edd28dc813265d91ca266378eac061598b7e3e3e9843fd","family_id":"qf-area-rectangle-dimensions","seed":202610066008,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"3\",\"7\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,7)×[0,3) has width 7 and height 3, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"21","explanation":"Multiply width by height to obtain 21 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0009","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a44acb427c56bb9f46e4c5776ae14913b97d65f79693e37ba7885028c5c6f7c8","family_id":"qf-area-rectangle-dimensions","seed":202610066009,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"4\",\"8\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,8)×[0,4) has width 8 and height 4, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"32","explanation":"Multiply width by height to obtain 32 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0010","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2069bd0895bbceb43fbf526aa5f0f7bdf34a706b1c2fc6d433d0a9ba0dba565e","family_id":"qf-area-rectangle-dimensions","seed":202610066010,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"5\",\"9\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,9)×[0,5) has width 9 and height 5, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"45","explanation":"Multiply width by height to obtain 45 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0011","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"757f8fa60e2ba34c5f8851404113e5f7adc0f8590c6a2e8e7a096e770ddc0373","family_id":"qf-area-rectangle-dimensions","seed":202610066011,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"1\",\"10\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,10)×[0,1) has width 10 and height 1, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"10","explanation":"Multiply width by height to obtain 10 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0012","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f45d1c1639b192f84a7e40244ec14a44116d2bde8a0b3f752cd3c07984b97359","family_id":"qf-area-rectangle-dimensions","seed":202610066012,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"2\",\"11\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,11)×[0,2) has width 11 and height 2, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"22","explanation":"Multiply width by height to obtain 22 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0013","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac894a7a3cdf24237dde5f6bfa0c73cc550c1e5e55064829165fa544f16fc2f6","family_id":"qf-area-rectangle-dimensions","seed":202610066013,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"3\",\"12\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,12)×[0,3) has width 12 and height 3, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"36","explanation":"Multiply width by height to obtain 36 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0014","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff665cbde23b20dcb0d422d23605ebb96b51faca7527da204b641850d500e0a5","family_id":"qf-area-rectangle-dimensions","seed":202610066014,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"4\",\"13\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,13)×[0,4) has width 13 and height 4, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"52","explanation":"Multiply width by height to obtain 52 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0015","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8b24405d772b19edc06b7774600b6f7df2de15ac0f8bc3e9b65e7b0a95d1a46","family_id":"qf-area-rectangle-dimensions","seed":202610066015,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"5\",\"14\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,14)×[0,5) has width 14 and height 5, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"70","explanation":"Multiply width by height to obtain 70 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0016","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3abe1d687488cf8e985855345870f13e8405a59d7e8dc5b5d871431cfaf7e40c","family_id":"qf-area-rectangle-dimensions","seed":202610066016,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"1\",\"15\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,15)×[0,1) has width 15 and height 1, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"15","explanation":"Multiply width by height to obtain 15 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0017","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e69366b9bd5d20f6d7be14493fac2e84303986dd575c3a4eb73b0b2ad87e900","family_id":"qf-area-rectangle-dimensions","seed":202610066017,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"2\",\"16\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,16)×[0,2) has width 16 and height 2, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"32","explanation":"Multiply width by height to obtain 32 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0018","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7f6c02dd1125d905a2dd4d7235f31e628a3c189cbc680d690f782c0b74e4266","family_id":"qf-area-rectangle-dimensions","seed":202610066018,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"3\",\"17\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,17)×[0,3) has width 17 and height 3, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"51","explanation":"Multiply width by height to obtain 51 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0019","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3610cd80c13736446b823c905b5b36a5b5d17ce2304ba2a08c2857047ce6f54","family_id":"qf-area-rectangle-dimensions","seed":202610066019,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"4\",\"18\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,18)×[0,4) has width 18 and height 4, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"72","explanation":"Multiply width by height to obtain 72 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0020","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9db962ebbf351d557916b160952db4e722f1ef72e1672ca9c75f0598b734bb6d","family_id":"qf-area-rectangle-dimensions","seed":202610066020,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"5\",\"19\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,19)×[0,5) has width 19 and height 5, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"95","explanation":"Multiply width by height to obtain 95 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0021","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ca273cab392a5b1136e8b8dac3efbcbece0c4fed4effc3d285aa5a15af04878","family_id":"qf-area-rectangle-dimensions","seed":202610066021,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"1\",\"20\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,20)×[0,1) has width 20 and height 1, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"20","explanation":"Multiply width by height to obtain 20 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0022","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48bde3fb6ff22ade3a02346a31042a9bacf5d55c34d59173b779a9821f2d411c","family_id":"qf-area-rectangle-dimensions","seed":202610066022,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"2\",\"21\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,21)×[0,2) has width 21 and height 2, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"42","explanation":"Multiply width by height to obtain 42 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0023","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0e4e0996b7b21d2d94471bed3a539ab73513dc583ba84931b5767777a906cfb","family_id":"qf-area-rectangle-dimensions","seed":202610066023,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"3\",\"22\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,22)×[0,3) has width 22 and height 3, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"66","explanation":"Multiply width by height to obtain 66 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0024","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e48d6097c3bf5e8f303d3b5c1d102e5cd33e51e342c81c0598115b2f27e40013","family_id":"qf-area-rectangle-dimensions","seed":202610066024,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"4\",\"23\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,23)×[0,4) has width 23 and height 4, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"92","explanation":"Multiply width by height to obtain 92 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0025","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d16573361036a283c6a93d45494ec76124bad44115495b8bb2acbd0e7ca92829","family_id":"qf-area-rectangle-dimensions","seed":202610066025,"semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"5\",\"24\"],\"operation\":\"rectangle_dimensions\"}"},"fields":{"title":"Area of a Half-Open Rectangle with Nonnegative Dimensions","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,24)×[0,5) has width 24 and height 5, both nonnegative. Find its area in squared length units.","answer":{"kind":"short_text","value":"120","explanation":"Multiply width by height to obtain 120 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter."},"skills":["Area of a Half-Open Rectangle with Nonnegative Dimensions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0026","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0c98d2d03706e79640eede0423859c96253b498d74fe1dfcc0827d19007e996","family_id":"qf-area-coordinate-rectangle","seed":202610066026,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"empty_rectangle\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [0, 0)×[-2, 3), find the exact area in squared length units.","answer":{"kind":"short_text","value":"0","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 0 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0027","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"670deeb181d7da6e6863734c4cc730767e8edc2df78e301009e188e0b2c3a74f","family_id":"qf-area-coordinate-rectangle","seed":202610066027,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"2\",\"2\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-5, -3)×[-11, -9), find the exact area in squared length units.","answer":{"kind":"short_text","value":"4","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 4 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0028","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce9cc5e9e793fa6cf8ab40cb841ad459c03102d47da4cdc33dd696d71c82bfba","family_id":"qf-area-coordinate-rectangle","seed":202610066028,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"3\",\"3\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-6, -3)×[-10, -7), find the exact area in squared length units.","answer":{"kind":"short_text","value":"9","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 9 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0029","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de63e798cec4bc67915258e6bc0f7de377d9c64079808a8cf74ab1da4a827396","family_id":"qf-area-coordinate-rectangle","seed":202610066029,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"4\",\"4\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-7, -3)×[-9, -5), find the exact area in squared length units.","answer":{"kind":"short_text","value":"16","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 16 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0030","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8448ae50ebfe662893e292f9f5fec0d6ce1eda1334e74a00b0f32e73df3165f3","family_id":"qf-area-coordinate-rectangle","seed":202610066030,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"1\",\"5\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-8, -7)×[-8, -3), find the exact area in squared length units.","answer":{"kind":"short_text","value":"5","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 5 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0031","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a0af70e981b33ce32bcb2c1948bbc9a860cf0912aec59e0e1b4596e14ecbb1f","family_id":"qf-area-coordinate-rectangle","seed":202610066031,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"2\",\"6\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-9, -7)×[-7, -1), find the exact area in squared length units.","answer":{"kind":"short_text","value":"12","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 12 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0032","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2296669cde36c2f44cf677f5d2c2703ffbbb5877e3c88f01a8717c9be9e6c639","family_id":"qf-area-coordinate-rectangle","seed":202610066032,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"3\",\"7\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-10, -7)×[-6, 1), find the exact area in squared length units.","answer":{"kind":"short_text","value":"21","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 21 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0033","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ad2489e06320f52ad62513b2769db03abf99f0b019d5079b10ca726c8156bf5","family_id":"qf-area-coordinate-rectangle","seed":202610066033,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"4\",\"8\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-11, -7)×[-5, 3), find the exact area in squared length units.","answer":{"kind":"short_text","value":"32","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 32 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0034","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54ccf55ca88abe246286ed6b840210d935238bbcc6fdd2b963b5780f47759bf7","family_id":"qf-area-coordinate-rectangle","seed":202610066034,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"1\",\"9\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-12, -11)×[-4, 5), find the exact area in squared length units.","answer":{"kind":"short_text","value":"9","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 9 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0035","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2629df75952a4abe98f727fa56aa63101b2b97e0aeac5c537c130004e4397ed2","family_id":"qf-area-coordinate-rectangle","seed":202610066035,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"2\",\"10\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-13, -11)×[-3, 7), find the exact area in squared length units.","answer":{"kind":"short_text","value":"20","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 20 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0036","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec7a7925a29fecde1a05ecc6a6742d086d69a4193101193b2349cc9ce7894ccf","family_id":"qf-area-coordinate-rectangle","seed":202610066036,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"3\",\"11\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-14, -11)×[-2, 9), find the exact area in squared length units.","answer":{"kind":"short_text","value":"33","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 33 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0037","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89b3c04e21fc89902353584b768fab7bf03086141b1bb3ba9b8479cc1559bc93","family_id":"qf-area-coordinate-rectangle","seed":202610066037,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"4\",\"12\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-15, -11)×[-1, 11), find the exact area in squared length units.","answer":{"kind":"short_text","value":"48","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 48 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0038","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b766e0e4b1d8790960aba0dbc81de620d00f200677d8365dfbf7d6a7fe89fc59","family_id":"qf-area-coordinate-rectangle","seed":202610066038,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"1\",\"13\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-16, -15)×[0, 13), find the exact area in squared length units.","answer":{"kind":"short_text","value":"13","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 13 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0039","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35b47297d1fe37d14f892f764ab16c092d9944da6ab9b35dd0e25a882a9e56e8","family_id":"qf-area-coordinate-rectangle","seed":202610066039,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"2\",\"14\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-17, -15)×[1, 15), find the exact area in squared length units.","answer":{"kind":"short_text","value":"28","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 28 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0040","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12f8309d369d1bbe089987d1c9c4b5ea98629f7087c3d8f4b7d8d998f056e5d1","family_id":"qf-area-coordinate-rectangle","seed":202610066040,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"3\",\"15\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-18, -15)×[2, 17), find the exact area in squared length units.","answer":{"kind":"short_text","value":"45","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 45 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0041","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56f5fcce1d11f78e423d8762c3570ee6fff3d19648a701d7636b55d6cb5b7825","family_id":"qf-area-coordinate-rectangle","seed":202610066041,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"4\",\"16\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-19, -15)×[3, 19), find the exact area in squared length units.","answer":{"kind":"short_text","value":"64","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 64 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0042","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73bc583fc75325fd566ec2e0a0ccbfe45fdcbe551e5d9e34287c49c4e79e56cc","family_id":"qf-area-coordinate-rectangle","seed":202610066042,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"1\",\"17\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-20, -19)×[4, 21), find the exact area in squared length units.","answer":{"kind":"short_text","value":"17","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 17 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0043","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9c6af2e6a52a2335e49bb5ba66644dda6f421e16286f3790720e662d0f3227e","family_id":"qf-area-coordinate-rectangle","seed":202610066043,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"2\",\"18\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-21, -19)×[5, 23), find the exact area in squared length units.","answer":{"kind":"short_text","value":"36","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 36 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0044","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff95dee47a858e6eec1dea7571a3ca2510c5e8ffcca747e261946251c17f4ff2","family_id":"qf-area-coordinate-rectangle","seed":202610066044,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"3\",\"19\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-22, -19)×[6, 25), find the exact area in squared length units.","answer":{"kind":"short_text","value":"57","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 57 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0045","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4c042793509f1ccba2fe2650ca2728d71f0f534f06ada19f15ceece1a9d0eb7","family_id":"qf-area-coordinate-rectangle","seed":202610066045,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"4\",\"20\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-23, -19)×[7, 27), find the exact area in squared length units.","answer":{"kind":"short_text","value":"80","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 80 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0046","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8c695eaca8507b9dd9e7fb0e93d900afc7e3724a9521ff0118fa43630a3ce59","family_id":"qf-area-coordinate-rectangle","seed":202610066046,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"1\",\"21\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-24, -23)×[8, 29), find the exact area in squared length units.","answer":{"kind":"short_text","value":"21","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 21 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0047","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d989badd7826b5db51992a44001dcd0d4f322d5e2edb514ccb047e3b8bc5be4","family_id":"qf-area-coordinate-rectangle","seed":202610066047,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"2\",\"22\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-25, -23)×[9, 31), find the exact area in squared length units.","answer":{"kind":"short_text","value":"44","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 44 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0048","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7372683781a646ece15a481343ededd986b420d4ea8a364ae32f9e2250d41c87","family_id":"qf-area-coordinate-rectangle","seed":202610066048,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"3\",\"23\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-26, -23)×[10, 33), find the exact area in squared length units.","answer":{"kind":"short_text","value":"69","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 69 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0049","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66f67aaf38f0a1c578a30717e281ed468c46cf8941138d746db5d0d069ebaad4","family_id":"qf-area-coordinate-rectangle","seed":202610066049,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"4\",\"24\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-27, -23)×[11, 35), find the exact area in squared length units.","answer":{"kind":"short_text","value":"96","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 96 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0050","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da1a7cb125a0c6cc8d62cc12e3068da902a24b0af56ee299b67506ae1acf292b","family_id":"qf-area-coordinate-rectangle","seed":202610066050,"semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"1\",\"25\"],\"operation\":\"coordinate_rectangle\"}"},"fields":{"title":"Area of a Half-Open Integer-Endpoint Rectangle","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-28, -27)×[12, 37), find the exact area in squared length units.","answer":{"kind":"short_text","value":"25","explanation":"Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 25 squared units. A zero width or height makes the rectangle empty."},"skills":["Area of a Half-Open Integer-Endpoint Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0051","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69ea570d8861b539b1f20d4605d949ce8f7e5f9f3fef1337be8fd76ac6646e2b","family_id":"qf-area-unit-cell-union","seed":202610066051,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[0],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as []; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"0","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 0 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0052","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df50c392348617fbfd386188b7872e593f97a813f7a9684a6f26ff8ebde1531d","family_id":"qf-area-unit-cell-union","seed":202610066052,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[1],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"1","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 1 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0053","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2325ca28e83c11446354e3e12a930eec4616b791aa073f8c773fcee035ce91c","family_id":"qf-area-unit-cell-union","seed":202610066053,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[2],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"2","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 2 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0054","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1fbb7d1b8f7e5442a4b0ba7fd7f4399091ce4aef4e6c0cd6828ee64262f91d1","family_id":"qf-area-unit-cell-union","seed":202610066054,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[3],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"3","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 3 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0055","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb5137eabd480d70617f0473ad7104be30d969af613ad84ac156c742ecc2ef66","family_id":"qf-area-unit-cell-union","seed":202610066055,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[4],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (-2, -2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"4","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 4 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0056","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9a007f796ca0af3339ec9d145f067c181bd2f613d3ba47b4189c54b13c8db44","family_id":"qf-area-unit-cell-union","seed":202610066056,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[5],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"5","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 5 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0057","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6906bb3c83abfa5a1973f810335a7c88d3976d083fe30b92c03c22006e59de8b","family_id":"qf-area-unit-cell-union","seed":202610066057,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[6],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"6","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 6 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0058","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3a76d7bf4a806513d21e4ab04816e66bcb177e0411d6cface8d660f621310c7","family_id":"qf-area-unit-cell-union","seed":202610066058,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[7],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"7","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 7 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0059","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68015c04c7cbd571de4aeb0a4ae7a5ea789997b2886285bb177f41728f7f8a62","family_id":"qf-area-unit-cell-union","seed":202610066059,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[8],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (-2, -2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"8","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 8 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0060","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0556cbb4f731876187fd93baf40d810c859cf5cd6f2f49a4adfdad12915c6da","family_id":"qf-area-unit-cell-union","seed":202610066060,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[9],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"9","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 9 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0061","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a4cdba5dc865a5fb74d7bdcda48d33dee6f7f20d9281693c6153eb5402f36fc","family_id":"qf-area-unit-cell-union","seed":202610066061,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[10],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"10","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 10 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0062","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0fec78ca51a1d80524a899241ae478ad14fb97bf1d6aceae31811d3134700f1","family_id":"qf-area-unit-cell-union","seed":202610066062,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[11],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"11","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 11 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0063","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e4b74c437b5f10e0bb72ff86c7f131ad00394245acf18c7de1950f7952c303a","family_id":"qf-area-unit-cell-union","seed":202610066063,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[12],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (-2, -2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"12","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 12 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0064","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb7200194f3b288a8079163c83ba5e1801bfbd26e2e64f9504b7f77106d38ea5","family_id":"qf-area-unit-cell-union","seed":202610066064,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[13],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"13","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 13 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0065","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9a89da5268e5bb5a4be082c1d7667ae12553bbce8ba06d68f4e663e1f6f8ef5","family_id":"qf-area-unit-cell-union","seed":202610066065,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[14],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"14","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 14 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0066","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6bc8740a921109e957a227712d88cbec889e9b7b031c71f8642e8dd829f71ba","family_id":"qf-area-unit-cell-union","seed":202610066066,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[15],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"15","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 15 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0067","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c39cad9db2a07737605f9ec36707001112ec0397a5894a08e4bcefb176d8eba0","family_id":"qf-area-unit-cell-union","seed":202610066067,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[16],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0), (-2, 1), (-2, -2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"16","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 16 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0068","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"783b8b6124820c0165e1a05c3b57e7b6dc7b83f4134b4a0e0d7c3a11cf0d94e4","family_id":"qf-area-unit-cell-union","seed":202610066068,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[17],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0), (-2, 1), (-1, 1)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"17","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 17 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0069","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4941d729efc06d2742c947282497ca8fac8d4d4f955a87253fdb8722c4da1461","family_id":"qf-area-unit-cell-union","seed":202610066069,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[18],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0), (-2, 1), (-1, 1), (0, 1)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"18","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 18 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0070","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df73fd6b0555558a6873db9929f8b654c35d01071a2f8c461fd02586408ef63a","family_id":"qf-area-unit-cell-union","seed":202610066070,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[19],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0), (-2, 1), (-1, 1), (0, 1), (1, 1)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"19","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 19 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0071","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c43075ab08c309101b4615584b40aff38427e54ee7cd0b4041f2193442e0e87","family_id":"qf-area-unit-cell-union","seed":202610066071,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[20],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0), (-2, 1), (-1, 1), (0, 1), (1, 1), (2, 1), (-2, -2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"20","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 20 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0072","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c811414642652e377ed8f3b6c01f14010646c88732737a8b30d5e65e09191641","family_id":"qf-area-unit-cell-union","seed":202610066072,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[21],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0), (-2, 1), (-1, 1), (0, 1), (1, 1), (2, 1), (-2, 2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"21","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 21 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0073","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c75de4159af97b37bdfa98f8479c24d41390953523b81e0bc897c4b89ebf74d7","family_id":"qf-area-unit-cell-union","seed":202610066073,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[22],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0), (-2, 1), (-1, 1), (0, 1), (1, 1), (2, 1), (-2, 2), (-1, 2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"22","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 22 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0074","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4160666f003cc411da210a0a6a6b08bd7f8114b67c8b67f7896dcda5af22999","family_id":"qf-area-unit-cell-union","seed":202610066074,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[23],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0), (-2, 1), (-1, 1), (0, 1), (1, 1), (2, 1), (-2, 2), (-1, 2), (0, 2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"23","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 23 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0075","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ee0ec90e6012edd1f14a11c6c0e9b900e291e7fbdb7f29ae8cdd0d32d2ebfcf","family_id":"qf-area-unit-cell-union","seed":202610066075,"semantic_template":"qf-area-unit-cell-union/{\"inputs\":[24],\"operation\":\"unit_cell_union\"}"},"fields":{"title":"Area of a Half-Open Unit-Cell Union","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0), (-2, 1), (-1, 1), (0, 1), (1, 1), (2, 1), (-2, 2), (-1, 2), (0, 2), (1, 2), (-2, -2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.","answer":{"kind":"short_text","value":"24","explanation":"Deduplicate coordinate pairs and count the distinct unit cells once each, giving 24 squared units. Repeated listings add no area, and an empty list has area zero."},"skills":["Area of a Half-Open Unit-Cell Union"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0076","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1cacaba7fa214c6144b8a2551dc6f5ea1be6c800c6dbd9b9010e0cb7dd835731","family_id":"qf-area-two-rectangle-union","seed":202610066076,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[\"empty_union\"],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 0)×[0, 0) and B=[2, 2)×[1, 4) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"0","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 0 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0077","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fae85bd2cdc6258ae28e143df0d0a89e1b22499017d281637508feb951df24b0","family_id":"qf-area-two-rectangle-union","seed":202610066077,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"2\",\"0\",\"1\"],[\"4\",\"6\",\"0\",\"3\"]],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 2)×[0, 1) and B=[4, 6)×[0, 3) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"8","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 8 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0078","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d41d7dd6bec1efb058f20aee160b18f42ae0a82f5b2d165662daa5283bc6110","family_id":"qf-area-two-rectangle-union","seed":202610066078,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"3\",\"0\",\"2\"],[\"3\",\"5\",\"0\",\"2\"]],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 3)×[0, 2) and B=[3, 5)×[0, 2) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"10","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 10 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0079","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20b982b72d1c486d1cc5a083b30146455f01a1d8e7bc382b84f92b8460a4bb0c","family_id":"qf-area-two-rectangle-union","seed":202610066079,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"3\",\"0\",\"3\"],[\"1\",\"4\",\"1\",\"4\"]],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 3)×[0, 3) and B=[1, 4)×[1, 4) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"14","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 14 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0080","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"930441d6462936f36bb9a3225693082e252010c879d21a069629d7266c64f755","family_id":"qf-area-two-rectangle-union","seed":202610066080,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"4\",\"0\",\"3\"],[\"1\",\"3\",\"1\",\"2\"]],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 4)×[0, 3) and B=[1, 3)×[1, 2) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"12","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 12 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0081","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84bae3326ca89cd1223d0bd0dd2312b4990f348a4917dc8b62f239255e5d4635","family_id":"qf-area-two-rectangle-union","seed":202610066081,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"2\",\"0\",\"3\"],[\"0\",\"2\",\"0\",\"3\"]],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 2)×[0, 3) and B=[0, 2)×[0, 3) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"6","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 6 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0082","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a510cbff9706a636ab12a62ce28a4c7eea86c30b5d12472bebcfaecf83e3c895","family_id":"qf-area-two-rectangle-union","seed":202610066082,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"3\",\"0\",\"2\"]],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 0)×[0, 8) and B=[1, 4)×[2, 4) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"6","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 6 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0083","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3787a1fdd33bad07bdbe6a43465ebe234f4a0f7ba1526f829f52e0187f6d659e","family_id":"qf-area-two-rectangle-union","seed":202610066083,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"3\",\"0\",\"2\"],[\"5\",\"8\",\"0\",\"4\"]],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 3)×[0, 2) and B=[5, 8)×[0, 4) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"18","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 18 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0084","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fdcd51efa1fcde3816fa221b9dda70e4c0e9ca9bf9e431ed3954996bc43bfd88","family_id":"qf-area-two-rectangle-union","seed":202610066084,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"4\",\"0\",\"3\"],[\"4\",\"7\",\"0\",\"3\"]],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 4)×[0, 3) and B=[4, 7)×[0, 3) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"21","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 21 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0085","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9dd9c385b9034213bca66096e9c3b216cd6a1996993323a4965beebe8c22e6b3","family_id":"qf-area-two-rectangle-union","seed":202610066085,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"4\",\"0\",\"4\"],[\"1\",\"5\",\"1\",\"5\"]],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 4)×[0, 4) and B=[1, 5)×[1, 5) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"23","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 23 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0086","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"088cec271e0d7766df0ac00ee455ca9e25ccc44283b68575000b789840babc99","family_id":"qf-area-two-rectangle-union","seed":202610066086,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"5\",\"0\",\"4\"],[\"1\",\"4\",\"1\",\"3\"]],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 5)×[0, 4) and B=[1, 4)×[1, 3) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"20","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 20 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0087","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c9a052221f71b0236de85f53789ddb0f21e803afc2ff8568502282dc486fae5","family_id":"qf-area-two-rectangle-union","seed":202610066087,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"3\",\"0\",\"4\"],[\"0\",\"3\",\"0\",\"4\"]],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 3)×[0, 4) and B=[0, 3)×[0, 4) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"12","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 12 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0088","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be98cae3e1d3de8bc209119eae90952336659b8336745913390f1fdc4d8ce9b8","family_id":"qf-area-two-rectangle-union","seed":202610066088,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"4\",\"0\",\"3\"]],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 0)×[0, 9) and B=[1, 5)×[2, 5) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"12","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 12 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0089","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5842691ed2e2487c8b3585743b999b547bef6e8bf61c3faa86180cb64326cbe","family_id":"qf-area-two-rectangle-union","seed":202610066089,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"4\",\"0\",\"3\"],[\"6\",\"10\",\"0\",\"5\"]],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 4)×[0, 3) and B=[6, 10)×[0, 5) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"32","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 32 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0090","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2578e5cf02d56c3f7a1dbe351789db3f6cbe467d351c24850030e063b701bb59","family_id":"qf-area-two-rectangle-union","seed":202610066090,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"5\",\"0\",\"4\"],[\"5\",\"9\",\"0\",\"4\"]],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 5)×[0, 4) and B=[5, 9)×[0, 4) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"36","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 36 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0091","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ec6aaa6652ae3c8ab52b1bdd7ec766f918731b3b0c70284e3597427c66edccc","family_id":"qf-area-two-rectangle-union","seed":202610066091,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"5\",\"0\",\"5\"],[\"1\",\"6\",\"1\",\"6\"]],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 5)×[0, 5) and B=[1, 6)×[1, 6) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"34","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 34 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0092","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1bc259996c88a54a21c02b8a982ceefc1a41014ff3ff5e1ecd378ca4c60125df","family_id":"qf-area-two-rectangle-union","seed":202610066092,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"6\",\"0\",\"5\"],[\"1\",\"5\",\"1\",\"4\"]],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 6)×[0, 5) and B=[1, 5)×[1, 4) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"30","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 30 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0093","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"687aaa948f09663e8fb089ec3be3b59cd5d13cac6fc197a4b2a61ab1d83ef2dd","family_id":"qf-area-two-rectangle-union","seed":202610066093,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"4\",\"0\",\"5\"],[\"0\",\"4\",\"0\",\"5\"]],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 4)×[0, 5) and B=[0, 4)×[0, 5) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"20","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 20 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0094","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3db5444163582395082a0c23c683013de4150671262326f9ed3fd695e22c5003","family_id":"qf-area-two-rectangle-union","seed":202610066094,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"5\",\"0\",\"4\"]],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 0)×[0, 10) and B=[1, 6)×[2, 6) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"20","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 20 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0095","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f95f3ef71136cd6b7db2b6ad21b399fc7004de93edd0ea9cdcaeb839f04fbab","family_id":"qf-area-two-rectangle-union","seed":202610066095,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"5\",\"0\",\"4\"],[\"7\",\"12\",\"0\",\"6\"]],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 5)×[0, 4) and B=[7, 12)×[0, 6) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"50","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 50 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0096","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"711b3836732c6e62a10a93ccca488f1b01c389f40d3b60b5a48f3650cc6513f3","family_id":"qf-area-two-rectangle-union","seed":202610066096,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"6\",\"0\",\"5\"],[\"6\",\"11\",\"0\",\"5\"]],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 6)×[0, 5) and B=[6, 11)×[0, 5) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"55","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 55 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0097","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b731de336c128e84aae1d3680ba40dccd84870982a96393ecbf90fcc1d927f1","family_id":"qf-area-two-rectangle-union","seed":202610066097,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"6\",\"0\",\"6\"],[\"1\",\"7\",\"1\",\"7\"]],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 6)×[0, 6) and B=[1, 7)×[1, 7) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"47","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 47 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0098","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fa23c6ffccef242fbf2a2902a7f1a2882697258f779b2b86314b212af257d4c","family_id":"qf-area-two-rectangle-union","seed":202610066098,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"7\",\"0\",\"6\"],[\"1\",\"6\",\"1\",\"5\"]],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 7)×[0, 6) and B=[1, 6)×[1, 5) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"42","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 42 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0099","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"828cced9630dbe7e7726f3e94fcf87ffeb8dc0d0c105db982b9b891be7d73f72","family_id":"qf-area-two-rectangle-union","seed":202610066099,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"5\",\"0\",\"6\"],[\"0\",\"5\",\"0\",\"6\"]],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 5)×[0, 6) and B=[0, 5)×[0, 6) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"30","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 30 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-area-0100","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afc093bc8341a29d74fcf73edb8be8d66661331c571224085dca289c401da2ec","family_id":"qf-area-two-rectangle-union","seed":202610066100,"semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"6\",\"0\",\"5\"]],\"operation\":\"two_rectangle_union\"}"},"fields":{"title":"Area of a Union of Two Half-Open Integer Rectangles","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 0)×[0, 11) and B=[1, 7)×[2, 7) are half-open rectangles. Find area(A∪B) in squared length units.","answer":{"kind":"short_text","value":"30","explanation":"For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 30 squared units. Disjoint or merely touching rectangles have zero overlap area."},"skills":["Area of a Union of Two Half-Open Integer Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}