{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0001","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,0)×[0,3) 的寬 w=0、高 h=3。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"0","explanation_zh":"面積=0 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066001,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":0,"h":3,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"empty_rectangle\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0002","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,1)×[0,2) 的寬 w=1、高 h=2。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"2","explanation_zh":"面積=2 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066002,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":1,"h":2,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"1\",\"2\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0003","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,2)×[0,3) 的寬 w=2、高 h=3。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"6","explanation_zh":"面積=6 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066003,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":2,"h":3,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"2\",\"3\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0004","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,3)×[0,4) 的寬 w=3、高 h=4。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"12","explanation_zh":"面積=12 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066004,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":3,"h":4,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"3\",\"4\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0005","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,4)×[0,5) 的寬 w=4、高 h=5。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"20","explanation_zh":"面積=20 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066005,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":4,"h":5,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"4\",\"5\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0006","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,5)×[0,1) 的寬 w=5、高 h=1。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"5","explanation_zh":"面積=5 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066006,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":5,"h":1,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"1\",\"5\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0007","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,6)×[0,2) 的寬 w=6、高 h=2。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"12","explanation_zh":"面積=12 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066007,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":6,"h":2,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"2\",\"6\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0008","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,7)×[0,3) 的寬 w=7、高 h=3。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"21","explanation_zh":"面積=21 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066008,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":7,"h":3,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"3\",\"7\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0009","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,8)×[0,4) 的寬 w=8、高 h=4。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"32","explanation_zh":"面積=32 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066009,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":8,"h":4,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"4\",\"8\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0010","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,9)×[0,5) 的寬 w=9、高 h=5。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"45","explanation_zh":"面積=45 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066010,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":9,"h":5,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"5\",\"9\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0011","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,10)×[0,1) 的寬 w=10、高 h=1。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"10","explanation_zh":"面積=10 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066011,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":10,"h":1,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"1\",\"10\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0012","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,11)×[0,2) 的寬 w=11、高 h=2。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"22","explanation_zh":"面積=22 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066012,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":11,"h":2,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"2\",\"11\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0013","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,12)×[0,3) 的寬 w=12、高 h=3。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"36","explanation_zh":"面積=36 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066013,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":12,"h":3,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"3\",\"12\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0014","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,13)×[0,4) 的寬 w=13、高 h=4。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"52","explanation_zh":"面積=52 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066014,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":13,"h":4,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"4\",\"13\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0015","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,14)×[0,5) 的寬 w=14、高 h=5。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"70","explanation_zh":"面積=70 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066015,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":14,"h":5,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"5\",\"14\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0016","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,15)×[0,1) 的寬 w=15、高 h=1。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"15","explanation_zh":"面積=15 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066016,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":15,"h":1,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"1\",\"15\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0017","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,16)×[0,2) 的寬 w=16、高 h=2。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"32","explanation_zh":"面積=32 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066017,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":16,"h":2,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"2\",\"16\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0018","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,17)×[0,3) 的寬 w=17、高 h=3。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"51","explanation_zh":"面積=51 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066018,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":17,"h":3,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"3\",\"17\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0019","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,18)×[0,4) 的寬 w=18、高 h=4。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"72","explanation_zh":"面積=72 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066019,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":18,"h":4,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"4\",\"18\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0020","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,19)×[0,5) 的寬 w=19、高 h=5。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"95","explanation_zh":"面積=95 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066020,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":19,"h":5,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"5\",\"19\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0021","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,20)×[0,1) 的寬 w=20、高 h=1。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"20","explanation_zh":"面積=20 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066021,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":20,"h":1,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"1\",\"20\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0022","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,21)×[0,2) 的寬 w=21、高 h=2。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"42","explanation_zh":"面積=42 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066022,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":21,"h":2,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"2\",\"21\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0023","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,22)×[0,3) 的寬 w=22、高 h=3。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"66","explanation_zh":"面積=66 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066023,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":22,"h":3,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"3\",\"22\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0024","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,23)×[0,4) 的寬 w=23、高 h=4。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"92","explanation_zh":"面積=92 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066024,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":23,"h":4,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"4\",\"23\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0025","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"非負尺寸半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,24)×[0,5) 的寬 w=24、高 h=5。求面積的精確平方單位數，零邊長允許空矩形。","answer":{"kind":"short_text","value":"120","explanation_zh":"面積=120 平方單位。寬乘高，任一邊長零即為空、面積零；不可用周長相加代替面積。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["非負尺寸半開矩形的面積"]},"variant":{"family_id":"qf-area-rectangle-dimensions","seed":202610066025,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","w":24,"h":5,"operation":"rectangle_dimensions","semantic_template":"qf-area-rectangle-dimensions/{\"inputs\":[\"5\",\"24\"],\"operation\":\"rectangle_dimensions\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0026","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [0,0)×[-2,3) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"0","explanation_zh":"面積=0 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066026,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[0,0,-2,3],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"empty_rectangle\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0027","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [-5,-3)×[-11,-9) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"4","explanation_zh":"面積=4 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066027,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[-5,-3,-11,-9],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"2\",\"2\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0028","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [-6,-3)×[-10,-7) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"9","explanation_zh":"面積=9 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066028,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[-6,-3,-10,-7],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"3\",\"3\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0029","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [-7,-3)×[-9,-5) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"16","explanation_zh":"面積=16 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066029,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[-7,-3,-9,-5],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"4\",\"4\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0030","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [-8,-7)×[-8,-3) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"5","explanation_zh":"面積=5 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066030,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[-8,-7,-8,-3],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"1\",\"5\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0031","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [-9,-7)×[-7,-1) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"12","explanation_zh":"面積=12 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066031,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[-9,-7,-7,-1],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"2\",\"6\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0032","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [-10,-7)×[-6,1) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"21","explanation_zh":"面積=21 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066032,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[-10,-7,-6,1],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"3\",\"7\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0033","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [-11,-7)×[-5,3) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"32","explanation_zh":"面積=32 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066033,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[-11,-7,-5,3],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"4\",\"8\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0034","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [-12,-11)×[-4,5) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"9","explanation_zh":"面積=9 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066034,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[-12,-11,-4,5],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"1\",\"9\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0035","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [-13,-11)×[-3,7) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"20","explanation_zh":"面積=20 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066035,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[-13,-11,-3,7],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"2\",\"10\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0036","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [-14,-11)×[-2,9) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"33","explanation_zh":"面積=33 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066036,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[-14,-11,-2,9],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"3\",\"11\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0037","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [-15,-11)×[-1,11) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"48","explanation_zh":"面積=48 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066037,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[-15,-11,-1,11],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"4\",\"12\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0038","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [-16,-15)×[0,13) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"13","explanation_zh":"面積=13 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066038,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[-16,-15,0,13],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"1\",\"13\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0039","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [-17,-15)×[1,15) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"28","explanation_zh":"面積=28 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066039,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[-17,-15,1,15],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"2\",\"14\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0040","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [-18,-15)×[2,17) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"45","explanation_zh":"面積=45 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066040,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[-18,-15,2,17],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"3\",\"15\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0041","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [-19,-15)×[3,19) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"64","explanation_zh":"面積=64 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066041,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[-19,-15,3,19],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"4\",\"16\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0042","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [-20,-19)×[4,21) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"17","explanation_zh":"面積=17 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066042,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[-20,-19,4,21],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"1\",\"17\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0043","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [-21,-19)×[5,23) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"36","explanation_zh":"面積=36 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066043,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[-21,-19,5,23],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"2\",\"18\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0044","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [-22,-19)×[6,25) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"57","explanation_zh":"面積=57 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066044,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[-22,-19,6,25],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"3\",\"19\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0045","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [-23,-19)×[7,27) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"80","explanation_zh":"面積=80 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066045,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[-23,-19,7,27],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"4\",\"20\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0046","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [-24,-23)×[8,29) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"21","explanation_zh":"面積=21 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066046,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[-24,-23,8,29],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"1\",\"21\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0047","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [-25,-23)×[9,31) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"44","explanation_zh":"面積=44 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066047,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[-25,-23,9,31],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"2\",\"22\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0048","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [-26,-23)×[10,33) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"69","explanation_zh":"面積=69 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066048,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[-26,-23,10,33],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"3\",\"23\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0049","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [-27,-23)×[11,35) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"96","explanation_zh":"面積=96 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066049,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[-27,-23,11,35],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"4\",\"24\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0050","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"整數端點半開矩形的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 [-28,-27)×[12,37) 端點有序。求面積的精確平方單位數。","answer":{"kind":"short_text","value":"25","explanation_zh":"面積=25 平方單位。先取兩個非負邊長，再相乘；零邊長是空矩形，座標原點移動不增加面積。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["整數端點半開矩形的面積"]},"variant":{"family_id":"qf-area-coordinate-rectangle","seed":202610066050,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","rect":[-28,-27,12,37],"operation":"coordinate_rectangle","semantic_template":"qf-area-coordinate-rectangle/{\"inputs\":[\"1\",\"25\"],\"operation\":\"coordinate_rectangle\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0051","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"0","explanation_zh":"面積=0 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066051,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[0],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0052","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-2,-2]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"1","explanation_zh":"面積=1 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066052,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[[-2,-2]],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[1],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0053","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-2,-2],[-1,-2]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"2","explanation_zh":"面積=2 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066053,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[[-2,-2],[-1,-2]],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[2],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0054","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-2,-2],[-1,-2],[0,-2]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"3","explanation_zh":"面積=3 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066054,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[[-2,-2],[-1,-2],[0,-2]],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[3],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0055","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-2,-2],[-1,-2],[0,-2],[1,-2],[-2,-2]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"4","explanation_zh":"面積=4 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066055,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[[-2,-2],[-1,-2],[0,-2],[1,-2],[-2,-2]],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[4],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0056","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-2,-2],[-1,-2],[0,-2],[1,-2],[2,-2]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"5","explanation_zh":"面積=5 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066056,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[[-2,-2],[-1,-2],[0,-2],[1,-2],[2,-2]],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[5],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0057","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-2,-2],[-1,-2],[0,-2],[1,-2],[2,-2],[-2,-1]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"6","explanation_zh":"面積=6 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066057,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[[-2,-2],[-1,-2],[0,-2],[1,-2],[2,-2],[-2,-1]],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[6],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0058","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-2,-2],[-1,-2],[0,-2],[1,-2],[2,-2],[-2,-1],[-1,-1]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"7","explanation_zh":"面積=7 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066058,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[[-2,-2],[-1,-2],[0,-2],[1,-2],[2,-2],[-2,-1],[-1,-1]],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[7],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0059","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-2,-2],[-1,-2],[0,-2],[1,-2],[2,-2],[-2,-1],[-1,-1],[0,-1],[-2,-2]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"8","explanation_zh":"面積=8 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066059,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[[-2,-2],[-1,-2],[0,-2],[1,-2],[2,-2],[-2,-1],[-1,-1],[0,-1],[-2,-2]],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[8],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0060","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-2,-2],[-1,-2],[0,-2],[1,-2],[2,-2],[-2,-1],[-1,-1],[0,-1],[1,-1]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"9","explanation_zh":"面積=9 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066060,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[[-2,-2],[-1,-2],[0,-2],[1,-2],[2,-2],[-2,-1],[-1,-1],[0,-1],[1,-1]],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[9],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0061","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-2,-2],[-1,-2],[0,-2],[1,-2],[2,-2],[-2,-1],[-1,-1],[0,-1],[1,-1],[2,-1]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"10","explanation_zh":"面積=10 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066061,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[[-2,-2],[-1,-2],[0,-2],[1,-2],[2,-2],[-2,-1],[-1,-1],[0,-1],[1,-1],[2,-1]],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[10],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0062","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-2,-2],[-1,-2],[0,-2],[1,-2],[2,-2],[-2,-1],[-1,-1],[0,-1],[1,-1],[2,-1],[-2,0]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"11","explanation_zh":"面積=11 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066062,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[[-2,-2],[-1,-2],[0,-2],[1,-2],[2,-2],[-2,-1],[-1,-1],[0,-1],[1,-1],[2,-1],[-2,0]],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[11],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0063","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-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]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"12","explanation_zh":"面積=12 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066063,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[[-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]],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[12],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0064","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-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]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"13","explanation_zh":"面積=13 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066064,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[[-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]],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[13],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0065","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-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]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"14","explanation_zh":"面積=14 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066065,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[[-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]],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[14],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0066","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-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]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"15","explanation_zh":"面積=15 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066066,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[[-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]],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[15],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0067","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-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]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"16","explanation_zh":"面積=16 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066067,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[[-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]],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[16],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0068","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-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]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"17","explanation_zh":"面積=17 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066068,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[[-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]],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[17],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0069","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-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]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"18","explanation_zh":"面積=18 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066069,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[[-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]],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[18],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0070","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-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]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"19","explanation_zh":"面積=19 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066070,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[[-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]],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[19],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0071","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-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]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"20","explanation_zh":"面積=20 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066071,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[[-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]],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[20],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0072","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-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]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"21","explanation_zh":"面積=21 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066072,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[[-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]],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[21],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0073","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-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]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"22","explanation_zh":"面積=22 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066073,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[[-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]],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[22],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0074","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-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]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"23","explanation_zh":"面積=23 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066074,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[[-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]],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[23],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0075","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"半開單位格聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-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]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。","answer":{"kind":"short_text","value":"24","explanation_zh":"面積=24 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["半開單位格聯集的面積"]},"variant":{"family_id":"qf-area-unit-cell-union","seed":202610066075,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","cells":[[-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]],"operation":"unit_cell_union","semantic_template":"qf-area-unit-cell-union/{\"inputs\":[24],\"operation\":\"unit_cell_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0076","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,0)×[0,0)，B=[2,2)×[1,4)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"0","explanation_zh":"面積=0 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066076,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,0,0,0],"B":[2,2,1,4],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[\"empty_union\"],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0077","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,2)×[0,1)，B=[4,6)×[0,3)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"8","explanation_zh":"面積=8 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066077,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,2,0,1],"B":[4,6,0,3],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"2\",\"0\",\"1\"],[\"4\",\"6\",\"0\",\"3\"]],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0078","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,3)×[0,2)，B=[3,5)×[0,2)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"10","explanation_zh":"面積=10 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066078,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,3,0,2],"B":[3,5,0,2],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"3\",\"0\",\"2\"],[\"3\",\"5\",\"0\",\"2\"]],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0079","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,3)×[0,3)，B=[1,4)×[1,4)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"14","explanation_zh":"面積=14 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066079,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,3,0,3],"B":[1,4,1,4],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"3\",\"0\",\"3\"],[\"1\",\"4\",\"1\",\"4\"]],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0080","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,4)×[0,3)，B=[1,3)×[1,2)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"12","explanation_zh":"面積=12 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066080,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,4,0,3],"B":[1,3,1,2],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"4\",\"0\",\"3\"],[\"1\",\"3\",\"1\",\"2\"]],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0081","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,2)×[0,3)，B=[0,2)×[0,3)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"6","explanation_zh":"面積=6 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066081,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,2,0,3],"B":[0,2,0,3],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"2\",\"0\",\"3\"],[\"0\",\"2\",\"0\",\"3\"]],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0082","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,0)×[0,8)，B=[1,4)×[2,4)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"6","explanation_zh":"面積=6 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066082,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,0,0,8],"B":[1,4,2,4],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"3\",\"0\",\"2\"]],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0083","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,3)×[0,2)，B=[5,8)×[0,4)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"18","explanation_zh":"面積=18 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066083,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,3,0,2],"B":[5,8,0,4],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"3\",\"0\",\"2\"],[\"5\",\"8\",\"0\",\"4\"]],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0084","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,4)×[0,3)，B=[4,7)×[0,3)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"21","explanation_zh":"面積=21 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066084,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,4,0,3],"B":[4,7,0,3],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"4\",\"0\",\"3\"],[\"4\",\"7\",\"0\",\"3\"]],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0085","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,4)×[0,4)，B=[1,5)×[1,5)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"23","explanation_zh":"面積=23 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066085,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,4,0,4],"B":[1,5,1,5],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"4\",\"0\",\"4\"],[\"1\",\"5\",\"1\",\"5\"]],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0086","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,5)×[0,4)，B=[1,4)×[1,3)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"20","explanation_zh":"面積=20 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066086,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,5,0,4],"B":[1,4,1,3],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"5\",\"0\",\"4\"],[\"1\",\"4\",\"1\",\"3\"]],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0087","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,3)×[0,4)，B=[0,3)×[0,4)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"12","explanation_zh":"面積=12 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066087,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,3,0,4],"B":[0,3,0,4],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"3\",\"0\",\"4\"],[\"0\",\"3\",\"0\",\"4\"]],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0088","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,0)×[0,9)，B=[1,5)×[2,5)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"12","explanation_zh":"面積=12 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066088,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,0,0,9],"B":[1,5,2,5],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"4\",\"0\",\"3\"]],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0089","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,4)×[0,3)，B=[6,10)×[0,5)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"32","explanation_zh":"面積=32 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066089,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,4,0,3],"B":[6,10,0,5],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"4\",\"0\",\"3\"],[\"6\",\"10\",\"0\",\"5\"]],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0090","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,5)×[0,4)，B=[5,9)×[0,4)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"36","explanation_zh":"面積=36 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066090,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,5,0,4],"B":[5,9,0,4],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"5\",\"0\",\"4\"],[\"5\",\"9\",\"0\",\"4\"]],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0091","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,5)×[0,5)，B=[1,6)×[1,6)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"34","explanation_zh":"面積=34 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066091,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,5,0,5],"B":[1,6,1,6],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"5\",\"0\",\"5\"],[\"1\",\"6\",\"1\",\"6\"]],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0092","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,6)×[0,5)，B=[1,5)×[1,4)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"30","explanation_zh":"面積=30 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066092,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,6,0,5],"B":[1,5,1,4],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"6\",\"0\",\"5\"],[\"1\",\"5\",\"1\",\"4\"]],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0093","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,4)×[0,5)，B=[0,4)×[0,5)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"20","explanation_zh":"面積=20 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066093,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,4,0,5],"B":[0,4,0,5],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"4\",\"0\",\"5\"],[\"0\",\"4\",\"0\",\"5\"]],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0094","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,0)×[0,10)，B=[1,6)×[2,6)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"20","explanation_zh":"面積=20 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066094,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,0,0,10],"B":[1,6,2,6],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"5\",\"0\",\"4\"]],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0095","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,5)×[0,4)，B=[7,12)×[0,6)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"50","explanation_zh":"面積=50 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066095,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,5,0,4],"B":[7,12,0,6],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"5\",\"0\",\"4\"],[\"7\",\"12\",\"0\",\"6\"]],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0096","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,6)×[0,5)，B=[6,11)×[0,5)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"55","explanation_zh":"面積=55 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066096,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,6,0,5],"B":[6,11,0,5],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"6\",\"0\",\"5\"],[\"6\",\"11\",\"0\",\"5\"]],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0097","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,6)×[0,6)，B=[1,7)×[1,7)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"47","explanation_zh":"面積=47 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066097,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,6,0,6],"B":[1,7,1,7],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"6\",\"0\",\"6\"],[\"1\",\"7\",\"1\",\"7\"]],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0098","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,7)×[0,6)，B=[1,6)×[1,5)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"42","explanation_zh":"面積=42 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066098,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,7,0,6],"B":[1,6,1,5],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"7\",\"0\",\"6\"],[\"1\",\"6\",\"1\",\"5\"]],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0099","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,5)×[0,6)，B=[0,5)×[0,6)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"30","explanation_zh":"面積=30 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066099,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,5,0,6],"B":[0,5,0,6],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"5\",\"0\",\"6\"],[\"0\",\"5\",\"0\",\"6\"]],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r06-area-0100","batch_id":"batch-20261005-advance-20261006-r06-area-foundations","language":"zh-Hant","source_mko_ids":["mko-area"],"title_zh":"兩個整數半開矩形聯集的面積","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,0)×[0,11)，B=[1,7)×[2,7)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。","answer":{"kind":"short_text","value":"30","explanation_zh":"面積=30 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["兩個整數半開矩形聯集的面積"]},"variant":{"family_id":"qf-area-two-rectangle-union","seed":202610066100,"parameters":{"frame":"euclidean_orthonormal_equal_units","boundary":"half_open","A":[0,0,0,11],"B":[1,7,2,7],"operation":"two_rectangle_union","semantic_template":"qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"6\",\"0\",\"5\"]],\"operation\":\"two_rectangle_union\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}