{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r05-circle-0001","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-12, 5)，非負半徑 r=0，圓 C={P:d(P,O)=r}。P=(-12, 5) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"是","explanation_zh":"平方比較=(0, 0)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。 半徑零時明示退化為僅含圓心的點集，沒有內部，直徑為零。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065001,"parameters":{"O":[-12,5],"r":0,"P":[-12,5],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"0\"],\"0\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0002","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-11, 4)，非負半徑 r=0，圓 C={P:d(P,O)=r}。P=(-10, 4) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"否","explanation_zh":"平方比較=(1, 0)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。 半徑零時明示退化為僅含圓心的點集，沒有內部，直徑為零。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065002,"parameters":{"O":[-11,4],"r":0,"P":[-10,4],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"1\"],\"0\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0003","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-10, 3)，非負半徑 r=1，圓 C={P:d(P,O)=r}。P=(-8, 4) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"否","explanation_zh":"平方比較=(5, 1)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065003,"parameters":{"O":[-10,3],"r":1,"P":[-8,4],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"1\",\"2\"],\"1\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0004","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-9, 2)，非負半徑 r=2，圓 C={P:d(P,O)=r}。P=(-7, 2) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"是","explanation_zh":"平方比較=(4, 4)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065004,"parameters":{"O":[-9,2],"r":2,"P":[-7,2],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"2\"],\"2\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0005","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-8, 1)，非負半徑 r=3，圓 C={P:d(P,O)=r}。P=(-8, 3) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"否","explanation_zh":"平方比較=(4, 9)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065005,"parameters":{"O":[-8,1],"r":3,"P":[-8,3],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"2\"],\"3\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0006","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-7, 0)，非負半徑 r=4，圓 C={P:d(P,O)=r}。P=(-2, 1) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"否","explanation_zh":"平方比較=(26, 16)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065006,"parameters":{"O":[-7,0],"r":4,"P":[-2,1],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"1\",\"5\"],\"4\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0007","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-6, -1)，非負半徑 r=5，圓 C={P:d(P,O)=r}。P=(-1, -1) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"是","explanation_zh":"平方比較=(25, 25)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065007,"parameters":{"O":[-6,-1],"r":5,"P":[-1,-1],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"5\"],\"5\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0008","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-5, -2)，非負半徑 r=6，圓 C={P:d(P,O)=r}。P=(-5, 3) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"否","explanation_zh":"平方比較=(25, 36)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065008,"parameters":{"O":[-5,-2],"r":6,"P":[-5,3],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"5\"],\"6\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0009","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-4, -3)，非負半徑 r=7，圓 C={P:d(P,O)=r}。P=(4, -2) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"否","explanation_zh":"平方比較=(65, 49)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065009,"parameters":{"O":[-4,-3],"r":7,"P":[4,-2],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"1\",\"8\"],\"7\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0010","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-3, -4)，非負半徑 r=8，圓 C={P:d(P,O)=r}。P=(5, -4) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"是","explanation_zh":"平方比較=(64, 64)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065010,"parameters":{"O":[-3,-4],"r":8,"P":[5,-4],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"8\"],\"8\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0011","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-2, -5)，非負半徑 r=9，圓 C={P:d(P,O)=r}。P=(-2, 3) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"否","explanation_zh":"平方比較=(64, 81)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065011,"parameters":{"O":[-2,-5],"r":9,"P":[-2,3],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"8\"],\"9\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0012","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-1, -6)，非負半徑 r=10，圓 C={P:d(P,O)=r}。P=(10, -5) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"否","explanation_zh":"平方比較=(122, 100)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065012,"parameters":{"O":[-1,-6],"r":10,"P":[10,-5],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"1\",\"11\"],\"10\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0013","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(0, -7)，非負半徑 r=11，圓 C={P:d(P,O)=r}。P=(11, -7) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"是","explanation_zh":"平方比較=(121, 121)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065013,"parameters":{"O":[0,-7],"r":11,"P":[11,-7],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"11\"],\"11\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0014","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(1, -8)，非負半徑 r=12，圓 C={P:d(P,O)=r}。P=(1, 3) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"否","explanation_zh":"平方比較=(121, 144)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065014,"parameters":{"O":[1,-8],"r":12,"P":[1,3],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"11\"],\"12\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0015","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(2, -9)，非負半徑 r=13，圓 C={P:d(P,O)=r}。P=(16, -8) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"否","explanation_zh":"平方比較=(197, 169)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065015,"parameters":{"O":[2,-9],"r":13,"P":[16,-8],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"1\",\"14\"],\"13\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0016","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(3, -10)，非負半徑 r=14，圓 C={P:d(P,O)=r}。P=(17, -10) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"是","explanation_zh":"平方比較=(196, 196)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065016,"parameters":{"O":[3,-10],"r":14,"P":[17,-10],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"14\"],\"14\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0017","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(4, -11)，非負半徑 r=15，圓 C={P:d(P,O)=r}。P=(4, 3) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"否","explanation_zh":"平方比較=(196, 225)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065017,"parameters":{"O":[4,-11],"r":15,"P":[4,3],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"14\"],\"15\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0018","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(5, -12)，非負半徑 r=16，圓 C={P:d(P,O)=r}。P=(22, -11) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"否","explanation_zh":"平方比較=(290, 256)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065018,"parameters":{"O":[5,-12],"r":16,"P":[22,-11],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"1\",\"17\"],\"16\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0019","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(6, -13)，非負半徑 r=17，圓 C={P:d(P,O)=r}。P=(23, -13) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"是","explanation_zh":"平方比較=(289, 289)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065019,"parameters":{"O":[6,-13],"r":17,"P":[23,-13],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"17\"],\"17\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0020","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(7, -14)，非負半徑 r=18，圓 C={P:d(P,O)=r}。P=(7, 3) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"否","explanation_zh":"平方比較=(289, 324)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065020,"parameters":{"O":[7,-14],"r":18,"P":[7,3],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"17\"],\"18\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0021","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(8, -15)，非負半徑 r=19，圓 C={P:d(P,O)=r}。P=(28, -14) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"否","explanation_zh":"平方比較=(401, 361)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065021,"parameters":{"O":[8,-15],"r":19,"P":[28,-14],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"1\",\"20\"],\"19\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0022","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(9, -16)，非負半徑 r=20，圓 C={P:d(P,O)=r}。P=(29, -16) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"是","explanation_zh":"平方比較=(400, 400)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065022,"parameters":{"O":[9,-16],"r":20,"P":[29,-16],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"20\"],\"20\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0023","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(10, -17)，非負半徑 r=21，圓 C={P:d(P,O)=r}。P=(10, 3) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"否","explanation_zh":"平方比較=(400, 441)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065023,"parameters":{"O":[10,-17],"r":21,"P":[10,3],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"20\"],\"21\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0024","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(11, -18)，非負半徑 r=22，圓 C={P:d(P,O)=r}。P=(34, -17) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"否","explanation_zh":"平方比較=(530, 484)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065024,"parameters":{"O":[11,-18],"r":22,"P":[34,-17],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"1\",\"23\"],\"22\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0025","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓邊界的成員判定","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(12, -19)，非負半徑 r=23，圓 C={P:d(P,O)=r}。P=(35, -19) 是否為圓邊界成員？圓不是包含內部的圓盤。","answer":{"kind":"short_text","value":"是","explanation_zh":"平方比較=(529, 529)。只有距中心平方等於 r² 才在圓邊界；圓內並不是圓上的同義詞。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["圓邊界的成員判定"]},"variant":{"family_id":"qf-circle-membership","seed":202610065025,"parameters":{"O":[12,-19],"r":23,"P":[35,-19],"frame":"euclidean_orthonormal_equal_units","operation":"circle_membership","semantic_template":"qf-circle-membership/{\"inputs\":[[\"0\",\"23\"],\"23\"],\"operation\":\"circle_membership\"}"}},"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-r05-circle-0026","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-12, 5)，非負半徑 r=0，圓 C={P:d(P,O)=r}。將 P=(-12, 5) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"圓上","explanation_zh":"平方比較=(0, 0)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。 半徑零時明示退化為僅含圓心的點集，沒有內部，直徑為零。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065026,"parameters":{"O":[-12,5],"r":0,"P":[-12,5],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"0\"],\"0\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0027","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-11, 4)，非負半徑 r=0，圓 C={P:d(P,O)=r}。將 P=(-10, 4) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"外","explanation_zh":"平方比較=(1, 0)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。 半徑零時明示退化為僅含圓心的點集，沒有內部，直徑為零。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065027,"parameters":{"O":[-11,4],"r":0,"P":[-10,4],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"1\"],\"0\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0028","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-10, 3)，非負半徑 r=1，圓 C={P:d(P,O)=r}。將 P=(-8, 4) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"外","explanation_zh":"平方比較=(5, 1)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065028,"parameters":{"O":[-10,3],"r":1,"P":[-8,4],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"1\",\"2\"],\"1\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0029","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-9, 2)，非負半徑 r=2，圓 C={P:d(P,O)=r}。將 P=(-7, 2) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"圓上","explanation_zh":"平方比較=(4, 4)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065029,"parameters":{"O":[-9,2],"r":2,"P":[-7,2],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"2\"],\"2\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0030","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-8, 1)，非負半徑 r=3，圓 C={P:d(P,O)=r}。將 P=(-8, 3) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"內","explanation_zh":"平方比較=(4, 9)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065030,"parameters":{"O":[-8,1],"r":3,"P":[-8,3],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"2\"],\"3\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0031","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-7, 0)，非負半徑 r=4，圓 C={P:d(P,O)=r}。將 P=(-2, 1) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"外","explanation_zh":"平方比較=(26, 16)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065031,"parameters":{"O":[-7,0],"r":4,"P":[-2,1],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"1\",\"5\"],\"4\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0032","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-6, -1)，非負半徑 r=5，圓 C={P:d(P,O)=r}。將 P=(-1, -1) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"圓上","explanation_zh":"平方比較=(25, 25)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065032,"parameters":{"O":[-6,-1],"r":5,"P":[-1,-1],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"5\"],\"5\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0033","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-5, -2)，非負半徑 r=6，圓 C={P:d(P,O)=r}。將 P=(-5, 3) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"內","explanation_zh":"平方比較=(25, 36)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065033,"parameters":{"O":[-5,-2],"r":6,"P":[-5,3],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"5\"],\"6\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0034","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-4, -3)，非負半徑 r=7，圓 C={P:d(P,O)=r}。將 P=(4, -2) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"外","explanation_zh":"平方比較=(65, 49)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065034,"parameters":{"O":[-4,-3],"r":7,"P":[4,-2],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"1\",\"8\"],\"7\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0035","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-3, -4)，非負半徑 r=8，圓 C={P:d(P,O)=r}。將 P=(5, -4) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"圓上","explanation_zh":"平方比較=(64, 64)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065035,"parameters":{"O":[-3,-4],"r":8,"P":[5,-4],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"8\"],\"8\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0036","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-2, -5)，非負半徑 r=9，圓 C={P:d(P,O)=r}。將 P=(-2, 3) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"內","explanation_zh":"平方比較=(64, 81)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065036,"parameters":{"O":[-2,-5],"r":9,"P":[-2,3],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"8\"],\"9\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0037","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-1, -6)，非負半徑 r=10，圓 C={P:d(P,O)=r}。將 P=(10, -5) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"外","explanation_zh":"平方比較=(122, 100)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065037,"parameters":{"O":[-1,-6],"r":10,"P":[10,-5],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"1\",\"11\"],\"10\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0038","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(0, -7)，非負半徑 r=11，圓 C={P:d(P,O)=r}。將 P=(11, -7) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"圓上","explanation_zh":"平方比較=(121, 121)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065038,"parameters":{"O":[0,-7],"r":11,"P":[11,-7],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"11\"],\"11\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0039","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(1, -8)，非負半徑 r=12，圓 C={P:d(P,O)=r}。將 P=(1, 3) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"內","explanation_zh":"平方比較=(121, 144)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065039,"parameters":{"O":[1,-8],"r":12,"P":[1,3],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"11\"],\"12\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0040","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(2, -9)，非負半徑 r=13，圓 C={P:d(P,O)=r}。將 P=(16, -8) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"外","explanation_zh":"平方比較=(197, 169)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065040,"parameters":{"O":[2,-9],"r":13,"P":[16,-8],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"1\",\"14\"],\"13\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0041","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(3, -10)，非負半徑 r=14，圓 C={P:d(P,O)=r}。將 P=(17, -10) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"圓上","explanation_zh":"平方比較=(196, 196)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065041,"parameters":{"O":[3,-10],"r":14,"P":[17,-10],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"14\"],\"14\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0042","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(4, -11)，非負半徑 r=15，圓 C={P:d(P,O)=r}。將 P=(4, 3) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"內","explanation_zh":"平方比較=(196, 225)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065042,"parameters":{"O":[4,-11],"r":15,"P":[4,3],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"14\"],\"15\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0043","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(5, -12)，非負半徑 r=16，圓 C={P:d(P,O)=r}。將 P=(22, -11) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"外","explanation_zh":"平方比較=(290, 256)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065043,"parameters":{"O":[5,-12],"r":16,"P":[22,-11],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"1\",\"17\"],\"16\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0044","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(6, -13)，非負半徑 r=17，圓 C={P:d(P,O)=r}。將 P=(23, -13) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"圓上","explanation_zh":"平方比較=(289, 289)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065044,"parameters":{"O":[6,-13],"r":17,"P":[23,-13],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"17\"],\"17\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0045","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(7, -14)，非負半徑 r=18，圓 C={P:d(P,O)=r}。將 P=(7, 3) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"內","explanation_zh":"平方比較=(289, 324)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065045,"parameters":{"O":[7,-14],"r":18,"P":[7,3],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"17\"],\"18\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0046","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(8, -15)，非負半徑 r=19，圓 C={P:d(P,O)=r}。將 P=(28, -14) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"外","explanation_zh":"平方比較=(401, 361)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065046,"parameters":{"O":[8,-15],"r":19,"P":[28,-14],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"1\",\"20\"],\"19\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0047","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(9, -16)，非負半徑 r=20，圓 C={P:d(P,O)=r}。將 P=(29, -16) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"圓上","explanation_zh":"平方比較=(400, 400)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065047,"parameters":{"O":[9,-16],"r":20,"P":[29,-16],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"20\"],\"20\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0048","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(10, -17)，非負半徑 r=21，圓 C={P:d(P,O)=r}。將 P=(10, 3) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"內","explanation_zh":"平方比較=(400, 441)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065048,"parameters":{"O":[10,-17],"r":21,"P":[10,3],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"20\"],\"21\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0049","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(11, -18)，非負半徑 r=22，圓 C={P:d(P,O)=r}。將 P=(34, -17) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"外","explanation_zh":"平方比較=(530, 484)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065049,"parameters":{"O":[11,-18],"r":22,"P":[34,-17],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"1\",\"23\"],\"22\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0050","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"相對圓邊界的三種位置","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(12, -19)，非負半徑 r=23，圓 C={P:d(P,O)=r}。將 P=(35, -19) 分為內、圓上或外；以到圓心距離與半徑比較。","answer":{"kind":"short_text","value":"圓上","explanation_zh":"平方比較=(529, 529)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["相對圓邊界的三種位置"]},"variant":{"family_id":"qf-circle-position","seed":202610065050,"parameters":{"O":[12,-19],"r":23,"P":[35,-19],"frame":"euclidean_orthonormal_equal_units","operation":"circle_position","semantic_template":"qf-circle-position/{\"inputs\":[[\"0\",\"23\"],\"23\"],\"operation\":\"circle_position\"}"}},"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-r05-circle-0051","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=0。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"0","explanation_zh":"直徑=0。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。 半徑零時明示退化為僅含圓心的點集，沒有內部，直徑為零。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065051,"parameters":{"r":0,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"0\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0052","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=1。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"2","explanation_zh":"直徑=2。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065052,"parameters":{"r":1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"1\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0053","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=2。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"4","explanation_zh":"直徑=4。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065053,"parameters":{"r":2,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"2\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0054","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=3。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"6","explanation_zh":"直徑=6。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065054,"parameters":{"r":3,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"3\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0055","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=4。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"8","explanation_zh":"直徑=8。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065055,"parameters":{"r":4,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"4\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0056","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=5。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"10","explanation_zh":"直徑=10。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065056,"parameters":{"r":5,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"5\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0057","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=6。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"12","explanation_zh":"直徑=12。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065057,"parameters":{"r":6,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"6\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0058","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=7。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"14","explanation_zh":"直徑=14。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065058,"parameters":{"r":7,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"7\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0059","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=8。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"16","explanation_zh":"直徑=16。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065059,"parameters":{"r":8,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"8\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0060","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=9。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"18","explanation_zh":"直徑=18。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065060,"parameters":{"r":9,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"9\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0061","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=10。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"20","explanation_zh":"直徑=20。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065061,"parameters":{"r":10,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"10\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0062","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=11。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"22","explanation_zh":"直徑=22。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065062,"parameters":{"r":11,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"11\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0063","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=12。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"24","explanation_zh":"直徑=24。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065063,"parameters":{"r":12,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"12\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0064","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=13。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"26","explanation_zh":"直徑=26。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065064,"parameters":{"r":13,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"13\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0065","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=14。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"28","explanation_zh":"直徑=28。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065065,"parameters":{"r":14,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"14\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0066","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=15。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"30","explanation_zh":"直徑=30。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065066,"parameters":{"r":15,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"15\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0067","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=16。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"32","explanation_zh":"直徑=32。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065067,"parameters":{"r":16,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"16\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0068","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=17。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"34","explanation_zh":"直徑=34。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065068,"parameters":{"r":17,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"17\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0069","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=18。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"36","explanation_zh":"直徑=36。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065069,"parameters":{"r":18,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"18\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0070","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=19。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"38","explanation_zh":"直徑=38。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065070,"parameters":{"r":19,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"19\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0071","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=20。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"40","explanation_zh":"直徑=40。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065071,"parameters":{"r":20,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"20\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0072","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=21。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"42","explanation_zh":"直徑=42。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065072,"parameters":{"r":21,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"21\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0073","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=22。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"44","explanation_zh":"直徑=44。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065073,"parameters":{"r":22,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"22\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0074","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=23。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"46","explanation_zh":"直徑=46。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065074,"parameters":{"r":23,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"23\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0075","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"非負半徑的直徑長","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓的非負半徑 r=24。求直徑長，使用同一長度單位。","answer":{"kind":"short_text","value":"48","explanation_zh":"直徑=48。直徑長是半徑的兩倍，不依無關的圓心位置變化，也不是面積或周長。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["非負半徑的直徑長"]},"variant":{"family_id":"qf-circle-diameter","seed":202610065075,"parameters":{"r":24,"frame":"euclidean_orthonormal_equal_units","operation":"circle_diameter","semantic_template":"qf-circle-diameter/{\"inputs\":[\"24\"],\"operation\":\"circle_diameter\"}"}},"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-r05-circle-0076","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(0, 0)，非負半徑 r=0，圓 C={P:d(P,O)=r}。從 O 沿 y 軸的 負方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"軸向點=(0, 0)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。 半徑零時明示退化為僅含圓心的點集，沒有內部，直徑為零。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065076,"parameters":{"O":[0,0],"r":0,"axis":"y","sign":-1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"0\",\"0\"],\"zero_radius\"],\"operation\":\"circle_axial\"}"}},"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-r05-circle-0077","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-11, 2)，非負半徑 r=1，圓 C={P:d(P,O)=r}。從 O 沿 x 軸的 正方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(-10, 2)","explanation_zh":"軸向點=(-10, 2)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065077,"parameters":{"O":[-11,2],"r":1,"axis":"x","sign":1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"-11\",\"2\"],\"1\",\"x\",1],\"operation\":\"circle_axial\"}"}},"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-r05-circle-0078","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-10, 1)，非負半徑 r=2，圓 C={P:d(P,O)=r}。從 O 沿 y 軸的 正方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(-10, 3)","explanation_zh":"軸向點=(-10, 3)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065078,"parameters":{"O":[-10,1],"r":2,"axis":"y","sign":1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"-10\",\"1\"],\"2\",\"y\",1],\"operation\":\"circle_axial\"}"}},"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-r05-circle-0079","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-9, 0)，非負半徑 r=3，圓 C={P:d(P,O)=r}。從 O 沿 x 軸的 負方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(-12, 0)","explanation_zh":"軸向點=(-12, 0)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065079,"parameters":{"O":[-9,0],"r":3,"axis":"x","sign":-1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"-9\",\"0\"],\"3\",\"x\",-1],\"operation\":\"circle_axial\"}"}},"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-r05-circle-0080","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-8, -1)，非負半徑 r=4，圓 C={P:d(P,O)=r}。從 O 沿 y 軸的 正方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(-8, 3)","explanation_zh":"軸向點=(-8, 3)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065080,"parameters":{"O":[-8,-1],"r":4,"axis":"y","sign":1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"-8\",\"-1\"],\"4\",\"y\",1],\"operation\":\"circle_axial\"}"}},"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-r05-circle-0081","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-7, -2)，非負半徑 r=5，圓 C={P:d(P,O)=r}。從 O 沿 x 軸的 正方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(-2, -2)","explanation_zh":"軸向點=(-2, -2)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065081,"parameters":{"O":[-7,-2],"r":5,"axis":"x","sign":1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"-7\",\"-2\"],\"5\",\"x\",1],\"operation\":\"circle_axial\"}"}},"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-r05-circle-0082","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-6, -3)，非負半徑 r=6，圓 C={P:d(P,O)=r}。從 O 沿 y 軸的 負方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(-6, -9)","explanation_zh":"軸向點=(-6, -9)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065082,"parameters":{"O":[-6,-3],"r":6,"axis":"y","sign":-1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"-6\",\"-3\"],\"6\",\"y\",-1],\"operation\":\"circle_axial\"}"}},"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-r05-circle-0083","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-5, -4)，非負半徑 r=7，圓 C={P:d(P,O)=r}。從 O 沿 x 軸的 正方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(2, -4)","explanation_zh":"軸向點=(2, -4)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065083,"parameters":{"O":[-5,-4],"r":7,"axis":"x","sign":1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"-5\",\"-4\"],\"7\",\"x\",1],\"operation\":\"circle_axial\"}"}},"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-r05-circle-0084","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-4, -5)，非負半徑 r=8，圓 C={P:d(P,O)=r}。從 O 沿 y 軸的 正方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(-4, 3)","explanation_zh":"軸向點=(-4, 3)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065084,"parameters":{"O":[-4,-5],"r":8,"axis":"y","sign":1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"-4\",\"-5\"],\"8\",\"y\",1],\"operation\":\"circle_axial\"}"}},"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-r05-circle-0085","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-3, -6)，非負半徑 r=9，圓 C={P:d(P,O)=r}。從 O 沿 x 軸的 負方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(-12, -6)","explanation_zh":"軸向點=(-12, -6)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065085,"parameters":{"O":[-3,-6],"r":9,"axis":"x","sign":-1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"-3\",\"-6\"],\"9\",\"x\",-1],\"operation\":\"circle_axial\"}"}},"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-r05-circle-0086","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-2, -7)，非負半徑 r=10，圓 C={P:d(P,O)=r}。從 O 沿 y 軸的 正方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(-2, 3)","explanation_zh":"軸向點=(-2, 3)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065086,"parameters":{"O":[-2,-7],"r":10,"axis":"y","sign":1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"-2\",\"-7\"],\"10\",\"y\",1],\"operation\":\"circle_axial\"}"}},"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-r05-circle-0087","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-1, -8)，非負半徑 r=11，圓 C={P:d(P,O)=r}。從 O 沿 x 軸的 正方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(10, -8)","explanation_zh":"軸向點=(10, -8)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065087,"parameters":{"O":[-1,-8],"r":11,"axis":"x","sign":1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"-1\",\"-8\"],\"11\",\"x\",1],\"operation\":\"circle_axial\"}"}},"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-r05-circle-0088","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(0, -9)，非負半徑 r=12，圓 C={P:d(P,O)=r}。從 O 沿 y 軸的 負方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(0, -21)","explanation_zh":"軸向點=(0, -21)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065088,"parameters":{"O":[0,-9],"r":12,"axis":"y","sign":-1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"0\",\"-9\"],\"12\",\"y\",-1],\"operation\":\"circle_axial\"}"}},"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-r05-circle-0089","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(1, -10)，非負半徑 r=13，圓 C={P:d(P,O)=r}。從 O 沿 x 軸的 正方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(14, -10)","explanation_zh":"軸向點=(14, -10)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065089,"parameters":{"O":[1,-10],"r":13,"axis":"x","sign":1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"1\",\"-10\"],\"13\",\"x\",1],\"operation\":\"circle_axial\"}"}},"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-r05-circle-0090","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(2, -11)，非負半徑 r=14，圓 C={P:d(P,O)=r}。從 O 沿 y 軸的 正方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(2, 3)","explanation_zh":"軸向點=(2, 3)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065090,"parameters":{"O":[2,-11],"r":14,"axis":"y","sign":1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"2\",\"-11\"],\"14\",\"y\",1],\"operation\":\"circle_axial\"}"}},"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-r05-circle-0091","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(3, -12)，非負半徑 r=15，圓 C={P:d(P,O)=r}。從 O 沿 x 軸的 負方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(-12, -12)","explanation_zh":"軸向點=(-12, -12)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065091,"parameters":{"O":[3,-12],"r":15,"axis":"x","sign":-1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"3\",\"-12\"],\"15\",\"x\",-1],\"operation\":\"circle_axial\"}"}},"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-r05-circle-0092","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(4, -13)，非負半徑 r=16，圓 C={P:d(P,O)=r}。從 O 沿 y 軸的 正方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(4, 3)","explanation_zh":"軸向點=(4, 3)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065092,"parameters":{"O":[4,-13],"r":16,"axis":"y","sign":1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"4\",\"-13\"],\"16\",\"y\",1],\"operation\":\"circle_axial\"}"}},"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-r05-circle-0093","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(5, -14)，非負半徑 r=17，圓 C={P:d(P,O)=r}。從 O 沿 x 軸的 正方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(22, -14)","explanation_zh":"軸向點=(22, -14)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065093,"parameters":{"O":[5,-14],"r":17,"axis":"x","sign":1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"5\",\"-14\"],\"17\",\"x\",1],\"operation\":\"circle_axial\"}"}},"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-r05-circle-0094","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(6, -15)，非負半徑 r=18，圓 C={P:d(P,O)=r}。從 O 沿 y 軸的 負方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(6, -33)","explanation_zh":"軸向點=(6, -33)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065094,"parameters":{"O":[6,-15],"r":18,"axis":"y","sign":-1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"6\",\"-15\"],\"18\",\"y\",-1],\"operation\":\"circle_axial\"}"}},"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-r05-circle-0095","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(7, -16)，非負半徑 r=19，圓 C={P:d(P,O)=r}。從 O 沿 x 軸的 正方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(26, -16)","explanation_zh":"軸向點=(26, -16)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065095,"parameters":{"O":[7,-16],"r":19,"axis":"x","sign":1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"7\",\"-16\"],\"19\",\"x\",1],\"operation\":\"circle_axial\"}"}},"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-r05-circle-0096","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(8, -17)，非負半徑 r=20，圓 C={P:d(P,O)=r}。從 O 沿 y 軸的 正方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(8, 3)","explanation_zh":"軸向點=(8, 3)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065096,"parameters":{"O":[8,-17],"r":20,"axis":"y","sign":1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"8\",\"-17\"],\"20\",\"y\",1],\"operation\":\"circle_axial\"}"}},"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-r05-circle-0097","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(9, -18)，非負半徑 r=21，圓 C={P:d(P,O)=r}。從 O 沿 x 軸的 負方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(-12, -18)","explanation_zh":"軸向點=(-12, -18)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065097,"parameters":{"O":[9,-18],"r":21,"axis":"x","sign":-1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"9\",\"-18\"],\"21\",\"x\",-1],\"operation\":\"circle_axial\"}"}},"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-r05-circle-0098","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(10, -19)，非負半徑 r=22，圓 C={P:d(P,O)=r}。從 O 沿 y 軸的 正方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(10, 3)","explanation_zh":"軸向點=(10, 3)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065098,"parameters":{"O":[10,-19],"r":22,"axis":"y","sign":1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"10\",\"-19\"],\"22\",\"y\",1],\"operation\":\"circle_axial\"}"}},"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-r05-circle-0099","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(11, -20)，非負半徑 r=23，圓 C={P:d(P,O)=r}。從 O 沿 x 軸的 正方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(34, -20)","explanation_zh":"軸向點=(34, -20)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065099,"parameters":{"O":[11,-20],"r":23,"axis":"x","sign":1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"11\",\"-20\"],\"23\",\"x\",1],\"operation\":\"circle_axial\"}"}},"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-r05-circle-0100","batch_id":"batch-20261005-advance-20261006-r05-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"圓上的指定軸向座標點","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(12, -21)，非負半徑 r=24，圓 C={P:d(P,O)=r}。從 O 沿 y 軸的 負方向移動 r，求所得軸向點的座標。","answer":{"kind":"expression","value":"(12, -45)","explanation_zh":"軸向點=(12, -45)。只在指定分量加上帶方向的半徑，另一分量保留；座標差的平方和為 r²。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["圓上的指定軸向座標點"]},"variant":{"family_id":"qf-circle-axial-point","seed":202610065100,"parameters":{"O":[12,-21],"r":24,"axis":"y","sign":-1,"frame":"euclidean_orthonormal_equal_units","operation":"circle_axial","semantic_template":"qf-circle-axial-point/{\"inputs\":[[\"12\",\"-21\"],\"24\",\"y\",-1],\"operation\":\"circle_axial\"}"}},"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"}}