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94dc7dc | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 | {"id": "669c46_l", "alias": "comb_sum_binomial_mod_v1_1742523217_226", "problem": "Let $n = 24$. Let $S = \\sum_{k=n}^{433} \\binom{T}{k}$, where $T = \\sum_{k=1}^{29} k$. Compute the remainder when $S$ is divided by $10859$.", "answer": 0, "graph": "", "domain": "ALG", "secondary_domain": "COMB", "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "SUM_ARITHMETIC", "lemma_paths": ["SUM_ARITHMETIC"], "recipe_id": "eb34f0", "seed_template_id": "comb_sum_binomial_mod_v1", "ending_id": null, "olympiad_level": 2, "variant": "legacy_text", "parent_id": "669c46", "num_spawns": 0, "lemma_set": ["SUM_ARITHMETIC"], "num_lemmas": 1, "generation_time": 0.232, "created_at": "2026-02-08T02:56:18.486476Z", "verification": {"verified": false, "answer": 3627, "error": "answer=0 != expected=3627", "timestamp": "2026-02-08T02:56:18.718065Z"}, "problem_hash": "662be2", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 167, "completion_tokens": 32768}, "timestamp": "2026-02-23T19:19:31.872Z"}, {"id": 2, "model": "openai/gpt-oss-120b", "answer": 4722, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 167, "completion_tokens": 19825}, "timestamp": "2026-02-23T14:51:17.041Z"}, {"id": 3, "model": "Qwen/Qwen3-235B-A22B-Thinking-2507", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 123, "completion_tokens": 32768}, "timestamp": "2026-02-24T00:15:03.507Z"}, {"id": 4, "model": "NousResearch/Hermes-4-405B", "answer": 10858, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 115, "completion_tokens": 642}, "timestamp": "2026-02-10T15:52:23.647Z"}, {"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 1, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 101, "completion_tokens": 6032}, "timestamp": "2026-02-10T18:22:21.739Z"}, {"id": 8, "model": "mathstral", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 156, "completion_tokens": 434}, "timestamp": "2026-02-23T14:07:25.960Z"}, {"id": 10, "model": "qwen2-math:7b", "answer": 8, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 118, "completion_tokens": 1174}, "timestamp": "2026-02-27T15:06:28.409Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 166, "completion_tokens": 434}, "timestamp": "2026-02-11T19:54:37.890Z"}, {"id": 15, "model": "Qwen/Qwen3-Coder-480B-A35B-Instruct", "answer": 2380, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 121, "completion_tokens": 3472}, "timestamp": "2026-02-12T02:02:20.050Z"}, {"id": 16, "model": "Qwen/Qwen3-Next-80B-A3B-Thinking", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 123, "completion_tokens": 32768}, "timestamp": "2026-02-23T23:11:33.209Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct-fast", "answer": 0, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 135, "completion_tokens": 1273}, "timestamp": "2026-02-12T00:26:46.921Z"}, {"id": 29, "model": "Qwen/Qwen3-235B-A22B-Instruct-2507", "answer": 0, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 121, "completion_tokens": 12698}, "timestamp": "2026-02-23T19:24:22.761Z"}, {"id": 36, "model": "qwen2.5:3b-instruct", "answer": 0, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 121, "completion_tokens": 317}, "timestamp": "2026-04-18T10:00:54.643Z"}], "solution_status": 0, "lemma_applicability": {"SUM_ARITHMETIC": "ok", "POLY_PADIC_VAL_CONST": "no"}, "irt_difficulty": {"lo": 5.33, "mid": 6.77, "hi": 8.69}}
{"id": "b36752", "alias": "comb_sum_binomial_row_v1_1742523217_3855", "problem": "Let $d$ be a positive integer such that $1 \\le d \\le 2$ and $d$ divides $14$. Let $m$ be the largest such $d$. Compute $m^{16}$.", "answer": 65536, "graph": "graphs = [\n Graph(\n let={\n \"_n\": Const(14),\n \"n\": Const(16),\n \"result\": Pow(MaxOverSet(set=SolutionsSet(var=Var(\"d\"), condition=And(Geq(Var(\"d\"), Const(1)), Leq(Var(\"d\"), Const(2)), Divides(divisor=Var(\"d\"), dividend=Ref(\"_n\"))))), Ref(\"n\")),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "SUM", "evaluator_id": "sympy", "root_lemma": "MAX_DIVISOR", "lemma_paths": ["MAX_DIVISOR"], "recipe_id": "51757e", "seed_template_id": "comb_sum_binomial_row_v1", "ending_id": null, "olympiad_level": 2, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["MAX_DIVISOR"], "num_lemmas": 1, "generation_time": 0.001, "created_at": "2026-02-08T06:07:02.496366Z", "verification": {"verified": true, "answer": 65536, "timestamp": "2026-02-08T06:07:02.497621Z"}, "problem_hash": "9ce8e1", "license": "CC BY 4.0", "llm_solvers": [{"id": 8, "model": "mathstral", "answer": 1048576, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 103, "completion_tokens": 223}, "timestamp": "2026-02-15T17:04:11.991Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "answer": 65536, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 150, "completion_tokens": 152}, "timestamp": "2026-02-11T23:37:31.297Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 65536, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 123, "completion_tokens": 292}, "timestamp": "2026-02-15T07:44:34.415Z"}], "solution_status": 1, "lemma_applicability": {"MAX_DIVISOR": "ok", "C2": "no", "L3b": "no", "LTE_DIFF_P2": "no", "MOD_FACTORIAL": "no", "V7": "no"}, "irt_difficulty": {"lo": -9.14, "mid": -6.05, "hi": -3.73}}
{"id": "368e63", "alias": "antilemma_k3_v1_1440796553_739", "problem": "Compute the sum of $\\phi(d)$ over all positive divisors $d$ of $43278$, where $\\phi$ denotes Euler's totient function.", "answer": 43278, "graph": "graphs = [\n Graph(\n let={\n \"_n\": Const(43278),\n \"x\": SumOverDivisors(n=Ref(name='_n'), var='d', expr=EulerPhi(n=Var(name='d'))),\n },\n goal=Ref(\"x\"),\n )\n]", "domain": "NT", "secondary_domain": "COMB", "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "K3", "lemma_paths": ["K3"], "recipe_id": "54c41e", "seed_template_id": "antilemma_k3_v1", "ending_id": null, "olympiad_level": 2, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["K3"], "num_lemmas": 1, "generation_time": 0.001, "created_at": "2026-02-08T11:56:50.999386Z", "verification": {"verified": true, "answer": 43278, "timestamp": "2026-02-08T11:56:50.999913Z"}, "problem_hash": "4f3d1e", "license": "CC BY 4.0", "llm_solvers": [{"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 43278, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 77, "completion_tokens": 1108}, "timestamp": "2026-02-14T20:46:41.737Z"}, {"id": 8, "model": "mathstral", "answer": 43278, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 130, "completion_tokens": 448}, "timestamp": "2026-02-22T03:49:58.854Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "answer": 14424, "score": 1, "correct": {"strict": false, "boxed": false, "relaxed": true}, "usage": {"prompt_tokens": 139, "completion_tokens": 467}, "timestamp": "2026-02-12T04:47:50.640Z"}], "solution_status": 1, "lemma_applicability": {"K3": "ok", "COUNT_DIVISOR_COUNT": "no", "K16": "no", "LTE_DIFF_P2": "no", "MAX_PRIME_BELOW": "no", "MOD_MUL": "no"}, "irt_difficulty": {"lo": -6.96, "mid": -4.56, "hi": -1.46}}
{"id": "8e31f9", "alias": "antilemma_product_of_sums_v1_153355830_73", "problem": "Let $x$ be the product of two quantities. The first is the sum of all positive integers $k$ such that $3^k \\le 71586494964$. The second is the sum of all $k$ such that $(k, j)$ is an ordered pair of positive integers with $1 \\le k \\le 8$ and $1 \\le j \\le 9$. Compute $x$.", "answer": 81972, "graph": "graphs = [\n Graph(\n let={\n \"_n\": Const(3),\n \"x\": Mul(Summation(var=\"k\", start=Const(1), end=MaxOverSet(set=SolutionsSet(var=Var(\"k\"), condition=Leq(Pow(Ref(\"_n\"), Var(\"k\")), Const(71586494964)))), expr=Var(\"k\")), SumOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"k\"), Var(\"_j\")]), condition=Const(1), domain=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(8)), right=IntegerRange(start=Const(1), end=Const(9)))), expr=Var(\"k\")))),\n \"Q\": Ref(\"x\"),\n },\n goal=Ref(\"Q\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "MAX_VAL", "lemma_paths": ["MAX_VAL/SUM_ARITHMETIC", "PRODUCT_OF_SUMS"], "recipe_id": "a95604", "seed_template_id": "antilemma_product_of_sums_v1", "ending_id": null, "olympiad_level": 3, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["MAX_VAL", "PRODUCT_OF_SUMS", "SUM_ARITHMETIC"], "num_lemmas": 3, "generation_time": 0.002, "created_at": "2026-02-08T02:52:59.497921Z", "verification": {"verified": true, "answer": 81972, "timestamp": "2026-02-08T02:52:59.499942Z"}, "problem_hash": "556a24", "license": "CC BY 4.0", "llm_solvers": [{"id": 2, "model": "openai/gpt-oss-120b", "answer": 81972, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 201, "completion_tokens": 1450}, "timestamp": "2026-02-08T22:41:51.875Z"}, {"id": 8, "model": "mathstral", "answer": 2808, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 188, "completion_tokens": 523}, "timestamp": "2026-02-17T15:10:04.489Z"}], "solution_status": 1, "lemma_applicability": {"MAX_VAL": "ok", "PRODUCT_OF_SUMS": "ok", "SUM_ARITHMETIC": "ok_later", "K16": "same_pattern_wrong", "COUNT_DIVISOR_COUNT": "no", "COUNT_FIB_DIVISIBLE": "no", "POLY_PADIC_VAL_CONST": "no", "V1": "no", "V8": "no"}, "irt_difficulty": {"lo": 3.31, "mid": 6.77, "hi": 10.0}}
{"id": "134295", "alias": "comb_count_derangements_v1_1353956133_34", "problem": "Let $\\_n = 2$. Define $n$ to be the smallest integer $d$ such that $d \\geq \\_n$ and $d$ divides $5929$. Compute the number of derangements of $n$ elements, denoted $!n$.", "answer": 1854, "graph": "graphs = [\n Graph(\n let={\n \"_n\": Const(2),\n \"n\": MinOverSet(set=SolutionsSet(var=Var(\"d\"), condition=And(Geq(Var(\"d\"), Ref(\"_n\")), Divides(divisor=Var(\"d\"), dividend=Const(5929))))),\n \"result\": Subfactorial(arg=Ref(name='n')),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "NT", "secondary_domain": "COMB", "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "MIN_PRIME_FACTOR", "lemma_paths": ["MIN_PRIME_FACTOR"], "recipe_id": "bc3776", "seed_template_id": "comb_count_derangements_v1", "ending_id": null, "olympiad_level": 3, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["MIN_PRIME_FACTOR"], "num_lemmas": 1, "generation_time": 0.001, "created_at": "2026-02-08T11:16:36.022895Z", "verification": {"verified": true, "answer": 1854, "timestamp": "2026-02-08T11:16:36.023863Z"}, "problem_hash": "6eb805", "license": "CC BY 4.0", "llm_solvers": [{"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 1854, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 103, "completion_tokens": 1122}, "timestamp": "2026-02-14T11:26:34.891Z"}, {"id": 8, "model": "mathstral", "answer": 1855, "score": 1, "correct": {"strict": false, "boxed": false, "relaxed": true}, "usage": {"prompt_tokens": 152, "completion_tokens": 759}, "timestamp": "2026-02-21T15:40:55.764Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 161, "completion_tokens": 329}, "timestamp": "2026-02-12T03:45:47.626Z"}], "solution_status": 1, "lemma_applicability": {"MIN_PRIME_FACTOR": "ok", "DS2": "no", "LTE_SUM": "no", "MAX_VAL": "no", "MOD_MUL": "no", "MOD_SUB": "no"}, "irt_difficulty": {"lo": -5.14, "mid": 0.32, "hi": 6.51}}
{"id": "edc245", "alias": "diophantine_product_count_v1_1742523217_2323", "problem": "Let $k = 1260$ and $u = 178$. Compute the number of positive integers $x$ such that $1 \\leq x \\leq u$, $x$ divides $k$, and $\\frac{k}{x} \\leq u$.", "answer": 22, "graph": "graphs = [\n Graph(\n let={\n \"k\": Const(1260),\n \"upper\": Const(178),\n \"result\": CountOverSet(set=SolutionsSet(var=Var(\"x\"), condition=And(Geq(Var(\"x\"), Const(1)), Leq(Var(\"x\"), Ref(\"upper\")), Divides(divisor=Var(\"x\"), dividend=Ref(\"k\")), Leq(Div(Ref(\"k\"), Var(\"x\")), Ref(\"upper\"))))),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "MIN_PRIME_FACTOR", "lemma_paths": ["MAX_DIVISOR"], "recipe_id": "51757e", "seed_template_id": "diophantine_product_count_v1", "ending_id": null, "olympiad_level": 3, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["MAX_DIVISOR", "MIN_PRIME_FACTOR"], "num_lemmas": 2, "generation_time": 0.877, "created_at": "2026-02-08T04:42:36.501360Z", "verification": {"verified": true, "answer": 22, "timestamp": "2026-02-08T04:42:37.378340Z"}, "problem_hash": "8f61bd", "license": "CC BY 4.0", "llm_solvers": [{"id": 2, "model": "openai/gpt-oss-120b", "answer": 22, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 166, "completion_tokens": 2520}, "timestamp": "2026-02-11T21:44:19.248Z"}, {"id": 8, "model": "mathstral", "answer": 35, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 115, "completion_tokens": 835}, "timestamp": "2026-02-13T22:34:36.794Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 161, "completion_tokens": 505}, "timestamp": "2026-02-11T21:41:25.313Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 22, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 133, "completion_tokens": 713}, "timestamp": "2026-02-14T04:38:16.503Z"}], "solution_status": 1, "lemma_applicability": {"MAX_DIVISOR": "ok", "K16": "no", "LTE_DIFF": "no", "V1": "no", "V5": "no", "V8_SUM": "no"}, "irt_difficulty": {"lo": -5.92, "mid": -3.14, "hi": 0.27}}
{"id": "239f84_n", "alias": "comb_count_surjections_v1_1218484723_5459", "problem": "A teacher assigns 7 distinct tasks to 2 identical teams such that each team gets at least one task. Teams are indistinct, so swapping team assignments doesn't count as a new arrangement. In how many ways can the tasks be assigned?", "answer": 126, "graph": "", "domain": "COMB", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "SUM_GEOM", "lemma_paths": ["SUM_GEOM"], "recipe_id": "04214c", "seed_template_id": "comb_count_surjections_v1", "ending_id": null, "olympiad_level": 3, "variant": "narrative", "parent_id": "239f84", "lemma_set": ["SUM_GEOM"], "num_lemmas": 1, "generation_time": 0.001, "created_at": "2026-02-25T07:00:40.563321Z", "problem_hash": "63e10b", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "answer": 63, "score": 1, "correct": {"strict": false, "boxed": false, "relaxed": true}, "usage": {"prompt_tokens": 158, "completion_tokens": 479}, "timestamp": "2026-03-30T23:27:19.065Z"}, {"id": 2, "model": "openai/gpt-oss-120b", "answer": 63, "score": 1, "correct": {"strict": false, "boxed": false, "relaxed": true}, "usage": {"prompt_tokens": 158, "completion_tokens": 487}, "timestamp": "2026-05-03T14:26:38.354Z"}, {"id": 16, "model": "Qwen/Qwen3-Next-80B-A3B-Thinking", "answer": 63, "score": 1, "correct": {"strict": false, "boxed": false, "relaxed": true}, "usage": {"prompt_tokens": 108, "completion_tokens": 4173}, "timestamp": "2026-03-31T19:09:55.004Z"}, {"id": 36, "model": "qwen2.5:3b-32k", "answer": 126, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 106, "completion_tokens": 276}, "timestamp": "2026-04-23T05:23:54.358Z"}], "solution_status": 1, "lemma_applicability": {"SUM_GEOM": "ok", "C2": "no", "COUNT_INTEGER_RANGE": "no", "COUNT_SUM_EQUALS": "no", "V7": "no", "V8": "no"}, "irt_difficulty": {"lo": 3.8, "mid": 6.33, "hi": 9.49}}
{"id": "76ffdf", "alias": "nt_count_primes_v1_1918700295_3531", "problem": "Let $A$ be the number of ordered pairs of positive integers $(p, q)$ such that $pq = 18$, $\\gcd(p, q) = 1$, and $p < q$. Let $U = 32768$. Define $S$ to be the set of all prime numbers $n$ such that $A \\leq n \\leq U$. Compute the number of elements in $S$.", "answer": 3512, "graph": "graphs = [\n Graph(\n let={\n \"upper\": Const(32768),\n \"result\": CountOverSet(set=SolutionsSet(var=Var(\"n\"), condition=And(Geq(Var(\"n\"), CountOverSet(set=SolutionsSet(var=Var(\"p\"), condition=And(IsPositive(arg=Var(name='p')), Exists(var=Var(name='q'), condition=And(Eq(left=Mul(Var(name='p'), Var(name='q')), right=Const(value=18)), Eq(left=GCD(a=Var(name='p'), b=Var(name='q')), right=Const(value=1)), Lt(left=Var(name='p'), right=Var(name='q')))))))), Leq(Var(\"n\"), Ref(\"upper\")), IsPrime(Var(\"n\"))))),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "COPRIME_PAIRS", "lemma_paths": ["COPRIME_PAIRS"], "recipe_id": "2bb3aa", "seed_template_id": "nt_count_primes_v1", "ending_id": null, "olympiad_level": 3, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["COPRIME_PAIRS"], "num_lemmas": 1, "generation_time": 0.893, "created_at": "2026-02-08T08:41:30.773132Z", "verification": {"verified": true, "answer": 3512, "timestamp": "2026-02-08T08:41:31.666500Z"}, "problem_hash": "cdd6f8", "license": "CC BY 4.0", "llm_solvers": [{"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 3512, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 132, "completion_tokens": 1622}, "timestamp": "2026-02-13T21:09:40.619Z"}, {"id": 8, "model": "mathstral", "answer": 1092, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 185, "completion_tokens": 389}, "timestamp": "2026-02-20T19:36:41.958Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 193, "completion_tokens": 331}, "timestamp": "2026-02-12T02:09:23.182Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 3512, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 164, "completion_tokens": 535}, "timestamp": "2026-02-16T15:32:54.137Z"}], "solution_status": 1, "lemma_applicability": {"COPRIME_PAIRS": "ok", "C2": "no", "COUNT_COPRIME_GRID": "no", "K15": "no", "LTE_DIFF_P2": "no", "V3": "no"}, "irt_difficulty": {"lo": -5.55, "mid": -3.01, "hi": 0.32}}
{"id": "e5a610", "alias": "modular_sum_quadratic_residues_v1_601307018_4777", "problem": "Let $p$ be the minimum value of $|x - y|$ over all ordered pairs $(x, y)$ of positive integers such that $xy = 427812$. Compute $\\frac{p(p - 1)}{4}$.", "answer": 53015, "graph": "graphs = [\n Graph(\n let={\n \"_n\": Const(4),\n \"p\": MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"x\"), Var(\"y\")]), condition=And(IsPositive(arg=Var(name='x')), IsPositive(arg=Var(name='y')), Eq(Mul(Var(\"x\"), Var(\"y\")), Const(427812)))), expr=Abs(arg=Sub(left=Var(name='x'), right=Var(name='y'))))),\n \"result\": Div(Mul(Ref(\"p\"), Sub(Ref(\"p\"), Const(1))), Ref(\"_n\")),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "SUM", "evaluator_id": "sympy", "root_lemma": "B3_DIFF", "lemma_paths": ["B3_DIFF"], "recipe_id": "b47ea7", "seed_template_id": "modular_sum_quadratic_residues_v1", "ending_id": null, "olympiad_level": 3, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["B3_DIFF"], "num_lemmas": 1, "generation_time": 0.003, "created_at": "2026-03-10T05:28:08.716994Z", "verification": {"verified": true, "answer": 53015, "timestamp": "2026-03-10T05:28:08.720206Z"}, "problem_hash": "6485ea", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "answer": 6271268, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 161, "completion_tokens": 12232}, "timestamp": "2026-03-29T13:26:05.478Z"}, {"id": 2, "model": "openai/gpt-oss-120b", "answer": 53015, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 161, "completion_tokens": 2147}, "timestamp": "2026-03-30T17:05:15.487Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 2, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 127, "completion_tokens": 901}, "timestamp": "2026-04-19T13:59:03.496Z"}, {"id": 36, "model": "qwen2.5:3b-32k", "answer": 0, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 110, "completion_tokens": 765}, "timestamp": "2026-04-22T13:00:44.049Z"}], "solution_status": 1, "lemma_applicability": {"B3_DIFF": "ok", "COUNT_COPRIME_GRID": "no", "K13": "no", "K15": "no", "L3b": "no", "V1": "no"}, "irt_difficulty": {"lo": 1.19, "mid": 4.27, "hi": 6.7}}
{"id": "5bd25b", "alias": "nt_max_prime_below_v1_784195855_5515", "problem": "Let $S$ be the set of all positive integers $p$ for which there exists a positive integer $q$ such that $pq = 36$, $\\gcd(p, q) = 1$, and $p < q$. Let $m$ be the number of elements in $S$. Determine the largest prime number $n$ such that $m \\leq n \\leq 24025$.", "answer": 24023, "graph": "graphs = [\n Graph(\n let={\n \"upper\": Const(24025),\n \"result\": MaxOverSet(set=SolutionsSet(var=Var(\"n\"), condition=And(Geq(Var(\"n\"), CountOverSet(set=SolutionsSet(var=Var(\"p\"), condition=And(IsPositive(arg=Var(name='p')), Exists(var=Var(name='q'), condition=And(Eq(left=Mul(Var(name='p'), Var(name='q')), right=Const(value=36)), Eq(left=GCD(a=Var(name='p'), b=Var(name='q')), right=Const(value=1)), Lt(left=Var(name='p'), right=Var(name='q')))))))), Leq(Var(\"n\"), Ref(\"upper\")), IsPrime(Var(\"n\"))))),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "EXTREMUM", "evaluator_id": "sympy", "root_lemma": "COPRIME_PAIRS", "lemma_paths": ["COPRIME_PAIRS"], "recipe_id": "2bb3aa", "seed_template_id": "nt_max_prime_below_v1", "ending_id": null, "olympiad_level": 3, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["COPRIME_PAIRS"], "num_lemmas": 1, "generation_time": 0.573, "created_at": "2026-02-08T07:56:26.158785Z", "verification": {"verified": true, "answer": 24023, "timestamp": "2026-02-08T07:56:26.731547Z"}, "problem_hash": "03a24a", "license": "CC BY 4.0", "llm_solvers": [{"id": 8, "model": "mathstral", "answer": 23981, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 145, "completion_tokens": 347}, "timestamp": "2026-02-15T19:07:13.031Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "answer": 24023, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 190, "completion_tokens": 322}, "timestamp": "2026-02-12T01:25:06.047Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 23907, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 161, "completion_tokens": 1316}, "timestamp": "2026-02-16T05:51:20.766Z"}], "solution_status": 1, "lemma_applicability": {"COPRIME_PAIRS": "ok", "L3c": "no", "LTE_SUM": "no", "MAX_VAL": "no", "MOD_ADD": "no", "MOD_FACTORIAL": "no"}, "irt_difficulty": {"lo": -5.55, "mid": -3.01, "hi": 0.34}}
{"id": "0f059f", "alias": "geo_count_lattice_rect_v1_1915831931_3216", "problem": "Let $a = 81$ and $b = 77$. Define $S$ as the set of all lattice points $(x, y)$ such that $0 \\leq x \\leq 81$ and $0 \\leq y \\leq 77$. Compute the number of elements in $S$.", "answer": 6396, "graph": "graphs = [\n Graph(\n let={\n \"a\": Const(81),\n \"b\": Const(77),\n \"result\": LatticePointsRect(a=Ref(name='a'), b=Ref(name='b')),\n \"Q\": Abs(arg=Ref(name='result')),\n },\n goal=Ref(\"Q\"),\n )\n]", "domain": "GEOM", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "", "lemma_paths": [], "recipe_id": "", "seed_template_id": "geo_count_lattice_rect_v1", "ending_id": null, "olympiad_level": 3, "variant": "", "parent_id": "", "num_spawns": 0, "generation_time": 0.001, "created_at": "2026-02-08T17:25:44.089610Z", "verification": {"verified": true, "answer": 6396, "timestamp": "2026-02-08T17:25:44.090573Z"}, "problem_hash": "b401a1", "license": "CC BY 4.0", "llm_solvers": [{"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 6396, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 109, "completion_tokens": 492}, "timestamp": "2026-02-24T22:39:52.189Z"}, {"id": 8, "model": "mathstral", "answer": 6264, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 121, "completion_tokens": 533}, "timestamp": "2026-02-16T09:45:39.426Z"}, {"id": 10, "model": "qwen2-math:7b", "answer": 6396, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 123, "completion_tokens": 383}, "timestamp": "2026-03-01T02:41:46.133Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "answer": 6396, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 169, "completion_tokens": 242}, "timestamp": "2026-02-12T22:39:12.346Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 6396, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 143, "completion_tokens": 278}, "timestamp": "2026-02-26T01:37:49.654Z"}, {"id": 36, "model": "qwen2.5:3b-32k", "answer": 6396, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 126, "completion_tokens": 460}, "timestamp": "2026-04-21T03:10:35.140Z"}], "solution_status": 1, "irt_difficulty": {"lo": -8.52, "mid": -5.43, "hi": -3.2}}
{"id": "b5b959", "alias": "modular_count_residue_v1_168721529_2099", "problem": "Let $\\_n = 57863$ and $\\text{upper} = 50176$. Let $m$ be the largest prime number in the range from 2 to 8, inclusive. Let $\\text{result}$ be the number of positive integers $n$ such that $1 \\leq n \\leq \\text{upper}$ and $n \\equiv 1 \\pmod{m}$.\n\nCompute $17373 \\cdot \\text{result} \\bmod \\_n$.\n\nFind the value of this expression.", "answer": 8488, "graph": "graphs = [\n Graph(\n let={\n \"_n\": Const(57863),\n \"upper\": Const(50176),\n \"m\": MaxOverSet(set=SolutionsSet(var=Var(\"n\"), condition=And(Geq(Var(\"n\"), Const(2)), Leq(Var(\"n\"), Const(8)), IsPrime(Var(\"n\"))))),\n \"r\": Const(1),\n \"result\": CountOverSet(set=SolutionsSet(var=Var(\"n\"), condition=And(Geq(Var(\"n\"), Const(1)), Leq(Var(\"n\"), Ref(\"upper\")), Eq(Mod(value=Var(\"n\"), modulus=Ref(\"m\")), Ref(\"r\"))))),\n \"Q\": Mod(value=Mul(Const(17373), Ref(\"result\")), modulus=Ref(\"_n\")),\n },\n goal=Ref(\"Q\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "MAX_PRIME_BELOW", "lemma_paths": ["MAX_PRIME_BELOW"], "recipe_id": "dc3ad3", "seed_template_id": "modular_count_residue_v1", "ending_id": null, "olympiad_level": 3, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["MAX_PRIME_BELOW"], "num_lemmas": 1, "generation_time": 4.066, "created_at": "2026-02-08T14:07:03.754256Z", "verification": {"verified": true, "answer": 8488, "timestamp": "2026-02-08T14:07:07.820402Z"}, "problem_hash": "8a0463", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "answer": 8488, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 224, "completion_tokens": 3266}, "timestamp": "2026-02-10T02:04:59.070Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "answer": 49968, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 220, "completion_tokens": 477}, "timestamp": "2026-02-11T21:18:33.355Z"}], "solution_status": 1, "lemma_applicability": {"MAX_PRIME_BELOW": "ok", "K14": "no", "K16": "no", "LTE_DIFF_P2": "no", "V1": "no", "V5": "no"}, "irt_difficulty": {"lo": -7.09, "mid": -0.49, "hi": 6.1}}
{"id": "7db533_l", "alias": "comb_sum_binomial_mod_v1_1116507919_98", "problem": "Let $S$ be the set of all integers $t$ such that $12 \\leq t \\leq 491$ and there exist positive integers $a$ and $b$ with $1 \\leq a \\leq 10$, $1 \\leq b \\leq 63$, and $t = 5a + 7b$. Let $c$ be the number of elements in $S$. Compute\n$$\n\\sum_{k=118}^{452} \\binom{c}{k}.\n$$\nLet $s$ denote this sum. Compute the remainder when $s$ is divided by $10859$. Let $r$ be this remainder. Determine the value of $43 - r \\bmod 80075$, taken as the unique integer in $[0, 80075)$.", "answer": 43, "graph": "", "domain": "ALG", "secondary_domain": "COMB", "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "LIN_FORM", "lemma_paths": ["LIN_FORM"], "recipe_id": "7b2633", "seed_template_id": "comb_sum_binomial_mod_v1", "ending_id": null, "olympiad_level": 4, "variant": "legacy_text", "parent_id": "7db533", "num_spawns": 0, "lemma_set": ["LIN_FORM"], "num_lemmas": 1, "generation_time": 0.036, "created_at": "2026-02-08T02:25:32.834692Z", "verification": {"verified": false, "answer": 76504, "error": "answer=43 != expected=76504", "timestamp": "2026-02-08T02:25:32.870861Z"}, "problem_hash": "3b7910", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 282, "completion_tokens": 32768}, "timestamp": "2026-02-23T15:18:43.866Z"}, {"id": 2, "model": "openai/gpt-oss-120b", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 282, "completion_tokens": 32768}, "timestamp": "2026-02-23T13:21:55.619Z"}, {"id": 3, "model": "Qwen/Qwen3-235B-A22B-Thinking-2507", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 250, "completion_tokens": 32768}, "timestamp": "2026-02-23T17:12:17.536Z"}, {"id": 4, "model": "NousResearch/Hermes-4-405B", "answer": 10765, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 230, "completion_tokens": 557}, "timestamp": "2026-02-10T12:38:25.936Z"}, {"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 43, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 207, "completion_tokens": 8054}, "timestamp": "2026-02-09T15:28:45.160Z"}, {"id": 8, "model": "mathstral", "answer": 129, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 243, "completion_tokens": 1081}, "timestamp": "2026-02-10T20:01:39.584Z"}, {"id": 10, "model": "qwen2-math:7b", "answer": 77869, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 245, "completion_tokens": 1293}, "timestamp": "2026-02-10T23:31:59.645Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 285, "completion_tokens": 464}, "timestamp": "2026-02-11T19:30:41.014Z"}, {"id": 15, "model": "Qwen/Qwen3-Coder-480B-A35B-Instruct", "answer": 43, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 248, "completion_tokens": 2638}, "timestamp": "2026-02-11T23:37:27.006Z"}, {"id": 16, "model": "Qwen/Qwen3-Next-80B-A3B-Thinking", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 250, "completion_tokens": 32768}, "timestamp": "2026-02-23T22:08:06.680Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct-fast", "answer": 42, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 250, "completion_tokens": 1243}, "timestamp": "2026-02-11T23:35:45.014Z"}, {"id": 29, "model": "Qwen/Qwen3-235B-A22B-Instruct-2507", "answer": 43, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 248, "completion_tokens": 14021}, "timestamp": "2026-02-23T15:10:07.495Z"}, {"id": 36, "model": "qwen2.5:3b-instruct", "answer": 76691, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 248, "completion_tokens": 1187}, "timestamp": "2026-04-18T09:29:52.963Z"}], "solution_status": 0, "lemma_applicability": {"LIN_FORM": "ok", "POLY_PADIC_VAL_CONST": "no"}, "irt_difficulty": {"lo": 6.44, "mid": 8.24, "hi": 10.0}}
{"id": "c4b58b", "alias": "antilemma_sum_equals_v1_784195855_4506", "problem": "Let $x$ be the number of ordered pairs $(i,j)$ of positive integers such that $i + j = 69$ and $1 \\leq i, j \\leq 67$. Compute the value of\n$$\n\\sum_{n=1}^{|x|} \\tau(n),\n$$\nwhere $\\tau(n)$ denotes the number of positive divisors of $n$.", "answer": 292, "graph": "graphs = [\n Graph(\n let={\n \"x\": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"i\"), Var(\"j\")]), condition=Eq(Sum(Var(\"i\"), Var(\"j\")), Const(69)), domain=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(67)), right=IntegerRange(start=Const(1), end=Const(67))))),\n \"Q\": Summation(var=\"n\", start=Const(1), end=Abs(arg=Ref(name='x')), expr=NumDivisors(n=Var(\"n\"))),\n },\n goal=Ref(\"Q\"),\n )\n]", "domain": "COMB", "secondary_domain": "GEOM", "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "LIN_FORM", "lemma_paths": ["COUNT_SUM_EQUALS"], "recipe_id": "75ab0f", "seed_template_id": "antilemma_sum_equals_v1", "ending_id": null, "olympiad_level": 4, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["COUNT_SUM_EQUALS", "LIN_FORM"], "num_lemmas": 2, "generation_time": 0.019, "created_at": "2026-02-08T07:09:05.789518Z", "verification": {"verified": true, "answer": 292, "timestamp": "2026-02-08T07:09:05.808562Z"}, "problem_hash": "95e604", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "answer": 292, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 192, "completion_tokens": 2636}, "timestamp": "2026-02-24T07:36:33.137Z"}, {"id": 2, "model": "openai/gpt-oss-120b", "answer": 292, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 192, "completion_tokens": 1967}, "timestamp": "2026-02-24T02:43:43.891Z"}, {"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 292, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 122, "completion_tokens": 2091}, "timestamp": "2026-02-13T08:14:17.441Z"}, {"id": 8, "model": "mathstral", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 179, "completion_tokens": 513}, "timestamp": "2026-02-20T00:03:13.053Z"}, {"id": 10, "model": "qwen2-math:7b", "answer": 324, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 138, "completion_tokens": 409}, "timestamp": "2026-02-28T03:16:05.977Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 187, "completion_tokens": 430}, "timestamp": "2026-02-12T00:51:47.842Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 1720, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 160, "completion_tokens": 747}, "timestamp": "2026-02-15T22:40:33.300Z"}, {"id": 36, "model": "qwen2.5:3b-32k", "answer": 100, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 141, "completion_tokens": 1177}, "timestamp": "2026-04-18T17:34:58.855Z"}], "solution_status": 1, "lemma_applicability": {"COUNT_SUM_EQUALS": "ok", "C2": "no", "COUNT_CARTESIAN": "no", "COUNT_COPRIME_GRID": "no", "V7": "no", "V8_SUM": "no"}, "irt_difficulty": {"lo": -2.35, "mid": 1.22, "hi": 4.84}}
{"id": "ae2bbe", "alias": "nt_min_crt_v1_397696148_1572", "problem": "Let $m = 4$ and $k = 5$. Let $a = 3$ and $b = 4$. Define $S$ as the set of all integers $n$ such that $1 \\leq n \\leq 20$, $n \\equiv 3 \\pmod{4}$, and $n \\equiv 4 \\pmod{5}$. Let $r$ be the minimum element of $S$. Compute $r + \\phi(|r| + 1) + \\tau(|r| + 1)$, where $\\phi(n)$ denotes Euler's totient function and $\\tau(n)$ denotes the number of positive divisors of $n$.", "answer": 33, "graph": "graphs = [\n Graph(\n let={\n \"m\": Const(4),\n \"k\": Const(5),\n \"a\": Const(3),\n \"b\": Const(4),\n \"upper\": Const(20),\n \"result\": MinOverSet(set=SolutionsSet(var=Var(\"n\"), condition=And(Geq(Var(\"n\"), Const(1)), Leq(Var(\"n\"), Ref(\"upper\")), Eq(Mod(value=Var(\"n\"), modulus=Ref(\"m\")), Ref(\"a\")), Eq(Mod(value=Var(\"n\"), modulus=Ref(\"k\")), Ref(\"b\"))))),\n \"Q\": Sum(Ref(\"result\"), EulerPhi(n=Sum(Abs(arg=Ref(name='result')), Const(1))), NumDivisors(n=Sum(Abs(arg=Ref(name='result')), Const(1)))),\n },\n goal=Ref(\"Q\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "EXTREMUM", "evaluator_id": "sympy", "root_lemma": "MOBIUS_COPRIME", "lemma_paths": ["MOBIUS_COPRIME", "B3"], "recipe_id": "233389", "seed_template_id": "nt_min_crt_v1", "ending_id": null, "olympiad_level": 4, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["B3", "MOBIUS_COPRIME"], "num_lemmas": 2, "generation_time": 0.206, "created_at": "2026-02-08T12:38:57.903373Z", "verification": {"verified": true, "answer": 33, "timestamp": "2026-02-08T12:38:58.108934Z"}, "problem_hash": "7adedf", "license": "CC BY 4.0", "llm_solvers": [{"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 33, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 185, "completion_tokens": 939}, "timestamp": "2026-02-15T03:09:41.448Z"}, {"id": 8, "model": "mathstral", "answer": 26, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 238, "completion_tokens": 448}, "timestamp": "2026-02-22T10:11:35.759Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 246, "completion_tokens": 617}, "timestamp": "2026-02-12T05:26:36.388Z"}], "solution_status": 1, "lemma_applicability": {"B3": "ok", "MOBIUS_COPRIME": "ok", "K13": "no", "V5": "no", "V8": "no", "V8_SUM": "no", "VAL_SUM_EQ": "no"}, "irt_difficulty": {"lo": -5.14, "mid": 0.32, "hi": 6.51}}
{"id": "6b4079", "alias": "sequence_lucas_compute_v1_2051736721_1495", "problem": "Let $t$ be an integer such that $17 \\leq t \\leq 77$. Suppose there exist positive integers $a$ and $b$ with $1 \\leq a \\leq 6$ and $1 \\leq b \\leq 4$ such that $t = 6a + 10b + 1$. Let $n$ be the number of integers $t$ satisfying these conditions. Compute the $n$-th Lucas number.", "answer": 64079, "graph": "graphs = [\n Graph(\n let={\n \"n\": CountOverSet(set=SolutionsSet(var=Var(\"t\"), condition=Exists(var=Var(name='a'), condition=Exists(var=Var(name='b'), condition=And(Geq(left=Var(name='a'), right=Const(value=1)), Leq(left=Var(name='a'), right=Const(value=6)), Geq(left=Var(name='b'), right=Const(value=1)), Leq(left=Var(name='b'), right=Const(value=4)), Geq(left=Var(name='t'), right=Const(value=17)), Leq(left=Var(name='t'), right=Const(value=77)), Eq(left=Var(name='t'), right=Sum(Mul(Const(value=6), Var(name='a')), Mul(Const(value=10), Var(name='b')), Const(value=1)))))))),\n \"result\": Lucas(arg=Ref(name='n')),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "LIN_FORM", "lemma_paths": ["LIN_FORM"], "recipe_id": "7b2633", "seed_template_id": "sequence_lucas_compute_v1", "ending_id": null, "olympiad_level": 4, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["LIN_FORM"], "num_lemmas": 1, "generation_time": 0.001, "created_at": "2026-02-08T16:04:09.292880Z", "verification": {"verified": true, "answer": 64079, "timestamp": "2026-02-08T16:04:09.294295Z"}, "problem_hash": "719cdf", "license": "CC BY 4.0", "llm_solvers": [{"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 64079, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 141, "completion_tokens": 2621}, "timestamp": "2026-02-16T20:43:40.850Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 200, "completion_tokens": 535}, "timestamp": "2026-02-12T09:50:53.564Z"}], "solution_status": 1, "lemma_applicability": {"LIN_FORM": "ok", "COUNT_DIVISOR_COUNT": "no", "K13": "no", "K18": "no", "MOD_FACTORIAL": "no", "MOD_MUL": "no"}, "irt_difficulty": {"lo": -7.08, "mid": -0.29, "hi": 6.49}}
{"id": "b4a087_l", "alias": "geo_count_lattice_rect_v1_784195855_7193", "problem": "Let $a = 349$ and $b = 105$. Define $R$ to be the number of lattice points in the rectangle $[0, a] \\times [0, b]$, including the boundary. Let $r$ be the remainder when $|R|$ is divided by $11$. Compute the Bell number $B_r$, which counts the number of partitions of a set of $r$ elements. Find the value of $B_r$.", "answer": 1, "graph": "", "domain": "GEOM", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "", "lemma_paths": [], "recipe_id": "", "seed_template_id": "geo_count_lattice_rect_v1", "ending_id": null, "olympiad_level": 4, "variant": "legacy_text", "parent_id": "b4a087", "num_spawns": 0, "generation_time": 0.002, "created_at": "2026-02-08T09:08:25.147989Z", "verification": {"verified": false, "answer": 4140, "error": "answer=1 != expected=4140", "timestamp": "2026-02-08T09:08:25.149540Z"}, "problem_hash": "10ea69", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "answer": 4140, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 206, "completion_tokens": 1630}, "timestamp": "2026-02-24T10:34:52.718Z"}, {"id": 2, "model": "openai/gpt-oss-120b", "answer": 4140, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 206, "completion_tokens": 596}, "timestamp": "2026-02-24T05:25:49.574Z"}, {"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 4140, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 143, "completion_tokens": 926}, "timestamp": "2026-02-14T00:52:49.127Z"}, {"id": 8, "model": "mathstral", "answer": 1, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 194, "completion_tokens": 397}, "timestamp": "2026-02-21T01:20:13.085Z"}, {"id": 10, "model": "qwen2-math:7b", "answer": 4238, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 153, "completion_tokens": 662}, "timestamp": "2026-02-28T06:39:19.814Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "answer": 1430, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 202, "completion_tokens": 257}, "timestamp": "2026-02-12T02:35:10.920Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 4140, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 172, "completion_tokens": 826}, "timestamp": "2026-02-16T21:23:13.910Z"}, {"id": 36, "model": "qwen2.5:3b-32k", "answer": 1, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 156, "completion_tokens": 521}, "timestamp": "2026-04-20T10:58:48.757Z"}], "solution_status": 1, "irt_difficulty": {"lo": -3.84, "mid": -1.67, "hi": 1.32}}
{"id": "693ed7", "alias": "comb_catalan_compute_v1_238844314_1128", "problem": "Let $n = 10$. Define $C_n$ to be the $n$th Catalan number. Compute the remainder when $\\sum_{k=\\binom{10}{10}}^{|C_n|} \\tau(k)$ is divided by 96591, where $\\tau(k)$ denotes the number of positive divisors of $k$.", "answer": 69404, "graph": "graphs = [\n Graph(\n let={\n \"n\": Const(10),\n \"result\": Catalan(Ref(\"n\")),\n \"Q\": Mod(value=Summation(var=\"n\", start=Binom(n=Const(10), k=Const(10)), end=Abs(arg=Ref(name='result')), expr=NumDivisors(n=Var(\"n\"))), modulus=Const(96591)),\n },\n goal=Ref(\"Q\"),\n )\n]", "domain": "COMB", "secondary_domain": "NT", "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "ONE_BINOM_N", "lemma_paths": ["ONE_BINOM_N"], "recipe_id": "9c72e5", "seed_template_id": "comb_catalan_compute_v1", "ending_id": null, "olympiad_level": 4, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["ONE_BINOM_N"], "num_lemmas": 1, "generation_time": 0.002, "created_at": "2026-02-08T13:58:49.697610Z", "verification": {"verified": true, "answer": 69404, "timestamp": "2026-02-08T13:58:49.699359Z"}, "problem_hash": "b4b986", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "answer": 69406, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 182, "completion_tokens": 9914}, "timestamp": "2026-02-24T19:31:41.821Z"}, {"id": 2, "model": "openai/gpt-oss-120b", "answer": 69404, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 182, "completion_tokens": 18802}, "timestamp": "2026-02-24T16:24:52.437Z"}, {"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 0, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 114, "completion_tokens": 3969}, "timestamp": "2026-02-15T22:52:05.677Z"}, {"id": 8, "model": "mathstral", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 173, "completion_tokens": 411}, "timestamp": "2026-02-26T14:54:07.079Z"}, {"id": 10, "model": "qwen2-math:7b", "answer": 85723, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 130, "completion_tokens": 1161}, "timestamp": "2026-02-28T16:22:06.787Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 180, "completion_tokens": 395}, "timestamp": "2026-02-12T07:39:15.283Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 0, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 148, "completion_tokens": 810}, "timestamp": "2026-02-25T04:27:49.697Z"}, {"id": 36, "model": "qwen2.5:3b-32k", "answer": 203, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 133, "completion_tokens": 730}, "timestamp": "2026-04-20T19:06:08.908Z"}], "solution_status": 1, "lemma_applicability": {"ONE_BINOM_N": "ok", "C2": "no", "COUNT_CARTESIAN": "no", "COUNT_COPRIME_GRID": "no", "V7": "no", "V8": "no"}, "irt_difficulty": {"lo": 3.25, "mid": 5.68, "hi": 8.81}}
{"id": "e18c24", "alias": "sequence_fibonacci_compute_v1_2051736721_5323", "problem": "Let $S$ be the set of all positive integers $t$ such that there exist positive integers $a$ and $b$ with $1 \\leq a \\leq 4$, $1 \\leq b \\leq 5$, $17 \\leq t \\leq 48$, and $t = 5a + 4b + 8$. Let $n$ be the number of elements in $S$. Compute the $n$-th Fibonacci number.", "answer": 6765, "graph": "graphs = [\n Graph(\n let={\n \"n\": CountOverSet(set=SolutionsSet(var=Var(\"t\"), condition=Exists(var=Var(name='a'), condition=Exists(var=Var(name='b'), condition=And(Geq(left=Var(name='a'), right=Const(value=1)), Leq(left=Var(name='a'), right=Const(value=4)), Geq(left=Var(name='b'), right=Const(value=1)), Leq(left=Var(name='b'), right=Const(value=5)), Geq(left=Var(name='t'), right=Const(value=17)), Leq(left=Var(name='t'), right=Const(value=48)), Eq(left=Var(name='t'), right=Sum(Mul(Const(value=5), Var(name='a')), Mul(Const(value=4), Var(name='b')), Const(value=8)))))))),\n \"result\": Fibonacci(arg=Ref(name='n')),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "LIN_FORM", "lemma_paths": ["LIN_FORM"], "recipe_id": "7b2633", "seed_template_id": "sequence_fibonacci_compute_v1", "ending_id": null, "olympiad_level": 4, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["LIN_FORM"], "num_lemmas": 1, "generation_time": 0.003, "created_at": "2026-02-08T18:29:52.084530Z", "verification": {"verified": true, "answer": 6765, "timestamp": "2026-02-08T18:29:52.087171Z"}, "problem_hash": "a8eb0f", "license": "CC BY 4.0", "llm_solvers": [{"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 6765, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 142, "completion_tokens": 1681}, "timestamp": "2026-02-18T17:24:33.455Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 200, "completion_tokens": 509}, "timestamp": "2026-02-13T00:11:31.441Z"}], "solution_status": 1, "lemma_applicability": {"LIN_FORM": "ok", "COUNT_DIVISOR_COUNT": "no", "K15": "no", "MOD_SUB": "no", "V5": "no", "VAL_SUM_EQ": "no"}, "irt_difficulty": {"lo": -7.08, "mid": -0.29, "hi": 6.49}}
{"id": "62e00d", "alias": "comb_count_surjections_v1_124444284_4943", "problem": "Let $n$ be the number of ordered pairs $(i, j)$ such that $i \\in \\{1, 2\\}$ and $j \\in \\{1, 2, 3\\}$. Let $k$ be the number of ordered pairs $(i, j)$ of positive integers with $1 \\leq i, j \\leq 7$ such that $i + j = 9$. Compute $k! \\cdot S(n, k)$, where $S(n, k)$ denotes the number of ways to partition a set of $n$ elements into $k$ nonempty unlabeled subsets.", "answer": 720, "graph": "graphs = [\n Graph(\n let={\n \"_n\": Const(9),\n \"n\": CountOverSet(set=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(2)), right=IntegerRange(start=Const(1), end=Const(3)))),\n \"k\": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"i\"), Var(\"j\")]), condition=Eq(Sum(Var(\"i\"), Var(\"j\")), Ref(\"_n\")), domain=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(7)), right=IntegerRange(start=Const(1), end=Const(7))))),\n \"result\": Mul(Factorial(Ref(\"k\")), Stirling2(n=Ref(name='n'), k=Ref(name='k'))),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "COMB", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "COUNT_SUM_EQUALS", "lemma_paths": ["COUNT_SUM_EQUALS", "COUNT_CARTESIAN"], "recipe_id": "e4fc6a", "seed_template_id": "comb_count_surjections_v1", "ending_id": null, "olympiad_level": 4, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["COUNT_CARTESIAN", "COUNT_SUM_EQUALS"], "num_lemmas": 2, "generation_time": 0.012, "created_at": "2026-02-08T06:18:37.029703Z", "verification": {"verified": true, "answer": 720, "timestamp": "2026-02-08T06:18:37.041364Z"}, "problem_hash": "dee583", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "answer": 720, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 243, "completion_tokens": 606}, "timestamp": "2026-02-24T05:53:44.732Z"}, {"id": 2, "model": "openai/gpt-oss-120b", "answer": 720, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 243, "completion_tokens": 553}, "timestamp": "2026-02-24T00:48:46.592Z"}, {"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 720, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 173, "completion_tokens": 578}, "timestamp": "2026-02-12T22:35:23.236Z"}, {"id": 8, "model": "mathstral", "answer": 360, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 222, "completion_tokens": 298}, "timestamp": "2026-02-19T04:11:26.297Z"}, {"id": 10, "model": "qwen2-math:7b", "answer": 720, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 185, "completion_tokens": 479}, "timestamp": "2026-02-28T01:21:39.553Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "answer": 2016, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 229, "completion_tokens": 522}, "timestamp": "2026-02-11T23:53:36.760Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 540, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 209, "completion_tokens": 1040}, "timestamp": "2026-02-15T10:49:42.410Z"}, {"id": 36, "model": "qwen2.5:3b-32k", "answer": 1600, "score": 1, "correct": {"strict": false, "boxed": false, "relaxed": true}, "usage": {"prompt_tokens": 188, "completion_tokens": 930}, "timestamp": "2026-04-18T16:11:38.612Z"}], "solution_status": 1, "lemma_applicability": {"COUNT_CARTESIAN": "ok", "COUNT_SUM_EQUALS": "ok", "C2": "no", "COUNT_COPRIME_GRID": "no", "V7": "no", "V8": "no", "V8_SUM": "no"}, "irt_difficulty": {"lo": -3.84, "mid": -1.69, "hi": 1.09}}
{"id": "305140", "alias": "v1_endings_v1_1116507919_169", "problem": "Let $n = 65444$ and $m = 20387$. Let $v_p(n!)$ denote the largest integer $k$ such that $p^k$ divides $n!$. Compute $v_5(n!) - v_5(m!)$.", "answer": 11262, "graph": "graphs = [\n Graph(\n let={\n \"n_val\": Const(65444),\n \"m_val\": Const(20387),\n \"p_val\": Const(5),\n \"n_fact\": Factorial(Ref(\"n_val\")),\n \"m_fact\": Factorial(Ref(\"m_val\")),\n \"vp_n\": MaxKDivides(target=Ref(\"n_fact\"), base=Ref(\"p_val\")),\n \"vp_m\": MaxKDivides(target=Ref(\"m_fact\"), base=Ref(\"p_val\")),\n \"x\": Sub(Ref(\"vp_n\"), Ref(\"vp_m\")),\n },\n goal=Ref(\"x\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "V1", "lemma_paths": ["V1"], "recipe_id": "dae96f", "seed_template_id": "v1_endings_v1", "ending_id": null, "olympiad_level": 4, "variant": "", "parent_id": "", "lemma_set": ["V1"], "num_lemmas": 1, "generation_time": 0.0, "created_at": "2026-02-08T02:27:05.716027Z", "verification": {"verified": true, "answer": 11262, "timestamp": "2026-02-08T02:27:05.716430Z"}, "problem_hash": "60e3c7", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "answer": 11262, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 170, "completion_tokens": 1066}, "timestamp": "2026-02-08T19:09:12.477Z"}, {"id": 2, "model": "openai/gpt-oss-120b", "answer": 11262, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 166, "completion_tokens": 1336}, "timestamp": "2026-02-09T14:03:56.379Z"}, {"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 11262, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 102, "completion_tokens": 1415}, "timestamp": "2026-02-09T16:44:44.405Z"}, {"id": 8, "model": "mathstral", "answer": 11242, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 159, "completion_tokens": 498}, "timestamp": "2026-02-17T12:27:36.184Z"}, {"id": 36, "model": "qwen2.5:3b-32k", "answer": 11155, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 118, "completion_tokens": 1223}, "timestamp": "2026-05-03T11:25:19.241Z"}], "solution_status": 1, "lemma_applicability": {"V1": "ok", "K5": "same_pattern_wrong", "COUNT_FIB_DIVISIBLE": "no", "DS2": "no", "K13": "no", "K18": "no", "MOD_ADD": "no"}, "irt_difficulty": {"lo": -4.6, "mid": 0.19, "hi": 4.77}}
{"id": "d0440c", "alias": "modular_inverse_v1_124444284_8804", "problem": "Let $x$ be a positive integer such that $1 \\leq x \\leq 336$ and\n$$\n33x \\equiv 1 \\pmod{337}.\n$$\nDetermine the value of $x$.", "answer": 143, "graph": "graphs = [\n Graph(\n let={\n \"a\": Const(33),\n \"m\": Const(337),\n \"upper\": Const(336),\n \"result\": MinOverSet(set=SolutionsSet(var=Var(\"x\"), condition=And(Geq(Var(\"x\"), Const(1)), Leq(Var(\"x\"), Ref(\"upper\")), Eq(Mod(value=Mul(Ref(\"a\"), Var(\"x\")), modulus=Ref(\"m\")), Const(1))))),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "EXTREMUM", "evaluator_id": "sympy", "root_lemma": "MAX_PRIME_BELOW", "lemma_paths": ["MAX_PRIME_BELOW"], "recipe_id": "dc3ad3", "seed_template_id": "modular_inverse_v1", "ending_id": null, "olympiad_level": 4, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["MAX_PRIME_BELOW"], "num_lemmas": 1, "generation_time": 0.074, "created_at": "2026-02-08T11:55:05.123272Z", "verification": {"verified": true, "answer": 143, "timestamp": "2026-02-08T11:55:05.197335Z"}, "problem_hash": "2532f5", "license": "CC BY 4.0", "llm_solvers": [{"id": 8, "model": "mathstral", "answer": 53, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 106, "completion_tokens": 464}, "timestamp": "2026-02-16T03:26:53.427Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "answer": 143, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 154, "completion_tokens": 524}, "timestamp": "2026-02-12T04:46:05.490Z"}], "solution_status": 1, "lemma_applicability": {"MAX_PRIME_BELOW": "ok", "K13": "no", "K5": "no", "LTE_SUM": "no", "POLY_PADIC_VAL_CONST": "no", "V8": "no"}, "irt_difficulty": {"lo": -6.96, "mid": -4.56, "hi": -1.46}}
{"id": "bf6805", "alias": "comb_count_permutations_fixed_v1_1520064083_9210", "problem": "Let $n = 8$ and $m = 245$. Let $k$ be the smallest divisor of $245$ that is at least $2$. Define $r = \\binom{8}{k} \\cdot !(8 - k)$, where $!a$ denotes the number of derangements of $a$ elements. Let $Q = 24025 - r$. Compute $Q$.", "answer": 23913, "graph": "graphs = [\n Graph(\n let={\n \"_n\": Const(245),\n \"n\": Const(8),\n \"k\": MinOverSet(set=SolutionsSet(var=Var(\"d\"), condition=And(Geq(Var(\"d\"), Const(2)), Divides(divisor=Var(\"d\"), dividend=Ref(\"_n\"))))),\n \"result\": Mul(Binom(n=Ref(\"n\"), k=Ref(\"k\")), Subfactorial(arg=Sub(left=Ref(name='n'), right=Ref(name='k')))),\n \"_c\": Const(24025),\n \"Q\": Sub(Ref(\"_c\"), Ref(\"result\")),\n },\n goal=Ref(\"Q\"),\n )\n]", "domain": "NT", "secondary_domain": "COMB", "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "MIN_PRIME_FACTOR", "lemma_paths": ["MIN_PRIME_FACTOR"], "recipe_id": "bc3776", "seed_template_id": "comb_count_permutations_fixed_v1", "ending_id": null, "olympiad_level": 4, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["MIN_PRIME_FACTOR"], "num_lemmas": 1, "generation_time": 0.002, "created_at": "2026-02-08T10:36:10.221374Z", "verification": {"verified": true, "answer": 23913, "timestamp": "2026-02-08T10:36:10.222976Z"}, "problem_hash": "9379bd", "license": "CC BY 4.0", "llm_solvers": [{"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 23913, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 130, "completion_tokens": 596}, "timestamp": "2026-02-14T07:51:04.706Z"}, {"id": 8, "model": "mathstral", "answer": 23913, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 184, "completion_tokens": 514}, "timestamp": "2026-02-21T09:36:39.115Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "answer": 23689, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 193, "completion_tokens": 283}, "timestamp": "2026-02-12T03:21:37.988Z"}], "solution_status": 1, "lemma_applicability": {"MIN_PRIME_FACTOR": "ok", "COUNT_COPRIME_GRID": "no", "K14": "no", "L3b": "no", "MOD_ADD": "no", "V7": "no"}, "irt_difficulty": {"lo": -9.12, "mid": -6.02, "hi": -3.62}}
{"id": "e32f24", "alias": "alg_linear_system_2x2_v1_1218484723_602", "problem": "Let $N$ be the number of integers $t$ with $14 \\le t \\le 2540$ that can be expressed as $t = 4a + 10b$ for some integers $a, b$ with $1 \\le a \\le 95$, $1 \\le b \\le 216$. Let $R$ be the number of integers $v$ with $5 \\le v \\le 980$ such that $v = 5b^2$ for some integer $b$ with $1 \\le b \\le 14$. Define $\\det = 16 \\cdot 6 + 5R$, $S = 241174 \\cdot 6 + 7431914$, and $T = 16 \\cdot (-74319) + 5 \\cdot 241174$. Let $K = \\frac{S}{\\det} + \\frac{T}{\\det}$. Find the remainder when $N - K$ is divided by $68924$.", "answer": 55098, "graph": "graphs = [\n Graph(\n let={\n \"_m\": Const(14),\n \"_n\": CountOverSet(set=SolutionsSet(var=Var(\"v\"), condition=And(Geq(Var(\"v\"), Const(5)), Leq(Var(\"v\"), Const(980)), Exists(var=Tuple(elements=[Var(name='a'), Var(name='b')]), condition=And(Geq(left=Var(name='a'), right=Const(value=1)), Leq(left=Var(name='a'), right=Const(value=14)), Geq(left=Var(name='b'), right=Const(value=1)), Leq(left=Var(name='b'), right=Const(value=14)), Eq(left=Mul(Const(value=5), Pow(base=Var(name='b'), exp=Const(value=2))), right=Var(name='v'))))))),\n \"num_x\": Sub(Mul(Const(241174), Const(6)), Mul(Const(-74319), Ref(\"_m\"))),\n \"num_y\": Sub(Mul(Const(16), Const(-74319)), Mul(Const(-5), Const(241174))),\n \"det\": Sub(Mul(Const(16), Const(6)), Mul(Const(-5), Ref(\"_n\"))),\n \"result\": Sum(Div(Ref(\"num_x\"), Ref(\"det\")), Div(Ref(\"num_y\"), Ref(\"det\"))),\n \"_c\": CountOverSet(set=SolutionsSet(var=Var(\"t\"), condition=Exists(var=Var(name='a'), condition=Exists(var=Var(name='b'), condition=And(Geq(left=Var(name='a'), right=Const(value=1)), Leq(left=Var(name='a'), right=Const(value=95)), Geq(left=Var(name='b'), right=Const(value=1)), Leq(left=Var(name='b'), right=Const(value=216)), Geq(left=Var(name='t'), right=Const(value=14)), Leq(left=Var(name='t'), right=Const(value=2540)), Eq(left=Var(name='t'), right=Sum(Mul(Const(value=4), Var(name='a')), Mul(Const(value=10), Var(name='b'))))))))),\n \"Q\": Mod(value=Sub(Ref(\"_c\"), Ref(\"result\")), modulus=Const(68924)),\n },\n goal=Ref(\"Q\"),\n )\n]", "domain": "ALG", "secondary_domain": null, "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "QF_PSD_DISTINCT", "lemma_paths": ["QF_PSD_DISTINCT/LIN_FORM", "LIN_FORM"], "recipe_id": "34b188", "seed_template_id": "alg_linear_system_2x2_v1", "ending_id": "negation_mod", "olympiad_level": 5, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["LIN_FORM", "QF_PSD_DISTINCT"], "num_lemmas": 2, "generation_time": 0.004, "created_at": "2026-02-25T02:17:05.435591Z", "verification": {"verified": true, "answer": 55098, "timestamp": "2026-02-25T02:17:05.439940Z"}, "problem_hash": "abc6ae", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 324, "completion_tokens": 32768}, "timestamp": "2026-03-28T23:26:42.667Z"}, {"id": 2, "model": "openai/gpt-oss-120b", "answer": 42342, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 324, "completion_tokens": 5208}, "timestamp": "2026-04-18T18:14:47.988Z"}, {"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 16698, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 252, "completion_tokens": 6014}, "timestamp": "2026-04-19T04:29:41.775Z"}, {"id": 10, "model": "qwen2-math:7b", "answer": 19501, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 292, "completion_tokens": 1015}, "timestamp": "2026-03-29T01:09:19.577Z"}, {"id": 16, "model": "Qwen/Qwen3-Next-80B-A3B-Thinking", "answer": 16596, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 297, "completion_tokens": 32492}, "timestamp": "2026-03-29T02:00:59.250Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 2540, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 286, "completion_tokens": 1178}, "timestamp": "2026-04-19T00:16:59.657Z"}, {"id": 36, "model": "qwen2.5:3b-32k", "answer": 741, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 295, "completion_tokens": 1299}, "timestamp": "2026-04-21T08:12:16.728Z"}, {"id": 38, "model": "google/gemma-3-27b-it", "answer": 16598, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 291, "completion_tokens": 2933}, "timestamp": "2026-04-19T13:52:27.730Z"}], "solution_status": 0, "lemma_applicability": {"LIN_FORM": "ok", "QF_PSD_DISTINCT": "ok", "POLY_PADIC_VAL_CONST": "no"}, "irt_difficulty": {"lo": 5.81, "mid": 8.21, "hi": 10.0}}
{"id": "69f9f9", "alias": "sequence_lucas_compute_v1_1978505735_1087", "problem": "Let $m = 144$. Let $P$ be the set of all ordered pairs $(x, y)$ of positive integers such that $xy = m$. For each such pair, compute $x + y$, and let $s$ be the minimum value of $x + y$ over all such pairs. Let $n$ be the largest prime number such that $2 \\leq n \\leq s$. Let $L_n$ denote the $n$th Lucas number, defined by $L_1 = 1$, $L_2 = 3$, and $L_k = L_{k-1} + L_{k-2}$ for $k \\geq 3$. Compute the remainder when $62843 \\cdot L_n$ is divided by $91788$.", "answer": 85249, "graph": "graphs = [\n Graph(\n let={\n \"_m\": Const(144),\n \"_n\": Const(91788),\n \"n\": MaxOverSet(set=SolutionsSet(var=Var(\"n1\"), condition=And(Geq(Var(\"n1\"), Const(2)), Leq(Var(\"n1\"), MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"x\"), Var(\"y\")]), condition=And(IsPositive(arg=Var(name='x')), IsPositive(arg=Var(name='y')), Eq(Mul(Var(\"x\"), Var(\"y\")), Ref(\"_m\")))), expr=Sum(Var(\"x\"), Var(\"y\"))))), IsPrime(Var(\"n1\"))))),\n \"result\": Lucas(arg=Ref(name='n')),\n \"_c\": Const(62843),\n \"Q\": Mod(value=Mul(Ref(\"_c\"), Ref(\"result\")), modulus=Ref(\"_n\")),\n },\n goal=Ref(\"Q\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "B3", "lemma_paths": ["B3/MAX_PRIME_BELOW"], "recipe_id": "f253c0", "seed_template_id": "sequence_lucas_compute_v1", "ending_id": null, "olympiad_level": 5, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["B3", "MAX_PRIME_BELOW"], "num_lemmas": 2, "generation_time": 0.003, "created_at": "2026-02-08T15:49:10.761032Z", "verification": {"verified": true, "answer": 85249, "timestamp": "2026-02-08T15:49:10.764058Z"}, "problem_hash": "6b12c3", "license": "CC BY 4.0", "llm_solvers": [{"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 85249, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 208, "completion_tokens": 3873}, "timestamp": "2026-02-16T14:44:08.607Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 275, "completion_tokens": 551}, "timestamp": "2026-02-12T09:17:34.202Z"}], "solution_status": 1, "lemma_applicability": {"B3": "ok", "MAX_PRIME_BELOW": "ok_later", "K14": "no", "K18": "no", "LTE_DIFF_P2": "no", "MOD_MUL": "no", "MOD_SUB": "no"}, "irt_difficulty": {"lo": -7.08, "mid": -0.29, "hi": 6.49}}
{"id": "b37b9c", "alias": "nt_num_divisors_compute_v1_1978505735_7197", "problem": "Let $n$ be the number of positive integers $p$ for which there exists a positive integer $q$ such that $p < q$, $\\gcd(p, q) = 1$, and $pq = 147476648205636750$. Let $d$ be the number of positive divisors of $n$. Compute the remainder when $44121d$ is divided by $95497$.", "answer": 66477, "graph": "graphs = [\n Graph(\n let={\n \"_n\": Const(95497),\n \"n\": CountOverSet(set=SolutionsSet(var=Var(\"p\"), condition=And(IsPositive(arg=Var(name='p')), Exists(var=Var(name='q'), condition=And(Eq(left=Mul(Var(name='p'), Var(name='q')), right=Const(value=147476648205636750)), Eq(left=GCD(a=Var(name='p'), b=Var(name='q')), right=Const(value=1)), Lt(left=Var(name='p'), right=Var(name='q'))))))),\n \"result\": NumDivisors(n=Ref(\"n\")),\n \"Q\": Mod(value=Mul(Const(44121), Ref(\"result\")), modulus=Ref(\"_n\")),\n },\n goal=Ref(\"Q\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "COPRIME_PAIRS", "lemma_paths": ["COPRIME_PAIRS"], "recipe_id": "2bb3aa", "seed_template_id": "nt_num_divisors_compute_v1", "ending_id": null, "olympiad_level": 5, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["COPRIME_PAIRS"], "num_lemmas": 1, "generation_time": 0.003, "created_at": "2026-02-08T20:06:47.117908Z", "verification": {"verified": true, "answer": 66477, "timestamp": "2026-02-08T20:06:47.121375Z"}, "problem_hash": "8a2f3b", "license": "CC BY 4.0", "llm_solvers": [{"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 66477, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 129, "completion_tokens": 4497}, "timestamp": "2026-02-18T23:59:16.940Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 204, "completion_tokens": 430}, "timestamp": "2026-02-13T01:16:52.904Z"}], "solution_status": 1, "lemma_applicability": {"COPRIME_PAIRS": "ok", "COUNT_FIB_DIVISIBLE": "no", "K15": "no", "K5": "no", "MAX_PRIME_BELOW": "no", "MOD_ADD": "no"}, "irt_difficulty": {"lo": -7.08, "mid": -0.29, "hi": 6.49}}
{"id": "91e3ee", "alias": "comb_count_surjections_v1_1978505735_1052", "problem": "Let $T$ be the number of integers $t$ between 15 and 36, inclusive, that can be written in the form $6a + 9b$ where $a$ is a positive integer at most 3 and $b$ is a positive integer at most 2. Let $n$ be the number of ordered pairs $(i, j)$ with $1 \\leq i, j \\leq 5$ such that $i + j = T$. Let $k$ be the number of ordered pairs $(x_1, x_2)$ of positive odd integers such that $x_1 + x_2 = 8$. Compute $k! \\cdot S(n, k)$, where $S(n, k)$ is the Stirling number of the second kind.", "answer": 240, "graph": "graphs = [\n Graph(\n let={\n \"_m\": Const(8),\n \"_n\": CountOverSet(set=SolutionsSet(var=Var(\"t\"), condition=Exists(var=Var(name='a'), condition=Exists(var=Var(name='b'), condition=And(Geq(left=Var(name='a'), right=Const(value=1)), Leq(left=Var(name='a'), right=Const(value=3)), Geq(left=Var(name='b'), right=Const(value=1)), Leq(left=Var(name='b'), right=Const(value=2)), Geq(left=Var(name='t'), right=Const(value=15)), Leq(left=Var(name='t'), right=Const(value=36)), Eq(left=Var(name='t'), right=Sum(Mul(Const(value=6), Var(name='a')), Mul(Const(value=9), Var(name='b'))))))))),\n \"n\": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"i\"), Var(\"j\")]), condition=Eq(Sum(Var(\"i\"), Var(\"j\")), Ref(\"_n\")), domain=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(5)), right=IntegerRange(start=Const(1), end=Const(5))))),\n \"k\": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"x1\"), Var(\"x2\")]), condition=And(IsPositive(arg=Var(name='x1')), IsPositive(arg=Var(name='x2')), IsOdd(arg=Var(name='x1')), IsOdd(arg=Var(name='x2')), Eq(Sum(Var(\"x1\"), Var(\"x2\")), Ref(\"_m\"))))),\n \"result\": Mul(Factorial(Ref(\"k\")), Stirling2(n=Ref(name='n'), k=Ref(name='k'))),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "COMB", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "LIN_FORM", "lemma_paths": ["LIN_FORM/COUNT_SUM_EQUALS", "COMB1"], "recipe_id": "81e769", "seed_template_id": "comb_count_surjections_v1", "ending_id": null, "olympiad_level": 5, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["COMB1", "COUNT_SUM_EQUALS", "LIN_FORM"], "num_lemmas": 3, "generation_time": 0.014, "created_at": "2026-02-08T15:45:31.370683Z", "verification": {"verified": true, "answer": 240, "timestamp": "2026-02-08T15:45:31.384249Z"}, "problem_hash": "fd851c", "license": "CC BY 4.0", "llm_solvers": [{"id": 2, "model": "openai/gpt-oss-120b", "answer": 240, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 277, "completion_tokens": 1143}, "timestamp": "2026-02-24T18:33:31.636Z"}, {"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 240, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 209, "completion_tokens": 969}, "timestamp": "2026-02-16T13:05:58.434Z"}, {"id": 10, "model": "qwen2-math:7b", "answer": 6, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 221, "completion_tokens": 806}, "timestamp": "2026-02-28T19:17:03.051Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "answer": 6, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 269, "completion_tokens": 619}, "timestamp": "2026-02-12T09:07:58.650Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 240, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 243, "completion_tokens": 936}, "timestamp": "2026-02-25T09:44:09.407Z"}, {"id": 36, "model": "qwen2.5:3b-32k", "answer": 240, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 224, "completion_tokens": 752}, "timestamp": "2026-04-20T21:21:10.325Z"}], "solution_status": 1, "lemma_applicability": {"COMB1": "ok", "LIN_FORM": "ok", "COUNT_SUM_EQUALS": "ok_later", "C2": "no", "COUNT_COPRIME_GRID": "no", "COUNT_INTEGER_RANGE": "no", "V8": "no", "V8_SUM": "no"}, "irt_difficulty": {"lo": -5.65, "mid": -3.17, "hi": -0.81}}
{"id": "556d55", "alias": "comb_count_surjections_v1_1526740231_167", "problem": "Let $n$ be the number of ordered pairs $(x_1, x_2)$ of positive odd integers such that $x_1 + x_2 = 14$. Let $k = 6$. Define $c = 75563$ and let $Q = c \\cdot k! \\cdot S(n, k)$, where $S(n, k)$ denotes the Stirling number of the second kind. Find the remainder when $Q$ is divided by $57001$.", "answer": 41517, "graph": "graphs = [\n Graph(\n let={\n \"n\": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"x1\"), Var(\"x2\")]), condition=And(IsPositive(arg=Var(name='x1')), IsPositive(arg=Var(name='x2')), IsOdd(arg=Var(name='x1')), IsOdd(arg=Var(name='x2')), Eq(Sum(Var(\"x1\"), Var(\"x2\")), Const(14))))),\n \"k\": Const(6),\n \"result\": Mul(Factorial(Ref(\"k\")), Stirling2(n=Ref(name='n'), k=Ref(name='k'))),\n \"_c\": Const(75563),\n \"Q\": Mod(value=Mul(Ref(\"_c\"), Ref(\"result\")), modulus=Const(57001)),\n },\n goal=Ref(\"Q\"),\n )\n]", "domain": "COMB", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "COMB1", "lemma_paths": ["COMB1"], "recipe_id": "567f58", "seed_template_id": "comb_count_surjections_v1", "ending_id": null, "olympiad_level": 5, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["COMB1"], "num_lemmas": 1, "generation_time": 0.002, "created_at": "2026-02-08T11:23:11.339097Z", "verification": {"verified": true, "answer": 41517, "timestamp": "2026-02-08T11:23:11.340938Z"}, "problem_hash": "87746b", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "answer": 41517, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 213, "completion_tokens": 2563}, "timestamp": "2026-02-24T13:37:07.619Z"}, {"id": 2, "model": "openai/gpt-oss-120b", "answer": 41517, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 213, "completion_tokens": 2271}, "timestamp": "2026-02-24T08:26:01.904Z"}, {"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 41517, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 146, "completion_tokens": 1891}, "timestamp": "2026-02-14T13:02:03.864Z"}, {"id": 8, "model": "mathstral", "answer": 29285, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 201, "completion_tokens": 1218}, "timestamp": "2026-02-21T18:04:08.261Z"}, {"id": 10, "model": "qwen2-math:7b", "answer": 29770, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 161, "completion_tokens": 2121}, "timestamp": "2026-02-28T09:05:24.667Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 211, "completion_tokens": 462}, "timestamp": "2026-02-12T03:56:16.247Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 39748, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 178, "completion_tokens": 703}, "timestamp": "2026-02-24T05:25:45.825Z"}, {"id": 36, "model": "qwen2.5:3b-32k", "answer": 0, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 164, "completion_tokens": 930}, "timestamp": "2026-04-20T13:08:17.250Z"}], "solution_status": 1, "lemma_applicability": {"COMB1": "ok", "COUNT_CARTESIAN": "no", "COUNT_INTEGER_RANGE": "no", "V7": "no", "V8": "no", "V8_SUM": "no"}, "irt_difficulty": {"lo": -2.35, "mid": 1.22, "hi": 4.84}}
{"id": "810ba0", "alias": "comb_catalan_compute_v1_784195855_5332", "problem": "Let $T$ be the number of ordered triples $(x_1, x_2, x_3)$ of positive odd integers such that $x_1 + x_2 + x_3 = 9$. Let $C_T$ denote the $T$-th Catalan number. Compute the remainder when $19909 \\cdot C_T$ is divided by 53481.", "answer": 28352, "graph": "graphs = [\n Graph(\n let={\n \"_n\": Const(53481),\n \"n\": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"x1\"), Var(\"x2\"), Var(\"x3\")]), condition=And(IsPositive(arg=Var(name='x1')), IsPositive(arg=Var(name='x2')), IsPositive(arg=Var(name='x3')), IsOdd(arg=Var(name='x1')), IsOdd(arg=Var(name='x2')), IsOdd(arg=Var(name='x3')), Eq(Sum(Var(\"x1\"), Var(\"x2\"), Var(\"x3\")), Const(9))))),\n \"result\": Catalan(Ref(\"n\")),\n \"Q\": Mod(value=Mul(Const(19909), Ref(\"result\")), modulus=Ref(\"_n\")),\n },\n goal=Ref(\"Q\"),\n )\n]", "domain": "COMB", "secondary_domain": null, "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "COMB1", "lemma_paths": ["COMB1"], "recipe_id": "567f58", "seed_template_id": "comb_catalan_compute_v1", "ending_id": null, "olympiad_level": 5, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["COMB1"], "num_lemmas": 1, "generation_time": 0.002, "created_at": "2026-02-08T07:49:43.771616Z", "verification": {"verified": true, "answer": 28352, "timestamp": "2026-02-08T07:49:43.773121Z"}, "problem_hash": "3850b1", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "answer": 28352, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 191, "completion_tokens": 2362}, "timestamp": "2026-02-24T08:29:18.002Z"}, {"id": 2, "model": "openai/gpt-oss-120b", "answer": 28352, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 191, "completion_tokens": 1418}, "timestamp": "2026-02-24T03:22:37.565Z"}, {"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 28352, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 126, "completion_tokens": 2250}, "timestamp": "2026-02-13T12:33:20.187Z"}, {"id": 8, "model": "mathstral", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 181, "completion_tokens": 706}, "timestamp": "2026-02-20T06:00:19.538Z"}, {"id": 10, "model": "qwen2-math:7b", "answer": 996, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 138, "completion_tokens": 857}, "timestamp": "2026-02-28T04:13:52.670Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "answer": 29990, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 189, "completion_tokens": 505}, "timestamp": "2026-02-12T01:19:27.124Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 44965, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 156, "completion_tokens": 675}, "timestamp": "2026-02-16T04:42:29.643Z"}, {"id": 36, "model": "qwen2.5:3b-32k", "answer": 53476, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 141, "completion_tokens": 989}, "timestamp": "2026-04-20T08:49:57.994Z"}], "solution_status": 1, "lemma_applicability": {"COMB1": "ok", "C2": "no", "COUNT_CARTESIAN": "no", "COUNT_COPRIME_GRID": "no", "COUNT_INTEGER_RANGE": "no", "COUNT_SUM_EQUALS": "no"}, "irt_difficulty": {"lo": -2.35, "mid": 1.22, "hi": 4.84}}
{"id": "4c236e", "alias": "geo_visible_lattice_v1_677425708_1010", "problem": "Let $n = 109$. Define a visible lattice point as an ordered pair of positive integers $(x, y)$ such that $1 \\le x, y \\le n$ and $\\gcd(x, y) = 1$. Compute the number of visible lattice points.", "answer": 7351, "graph": "graphs = [\n Graph(\n let={\n \"n\": Const(109),\n \"result\": VisibleLatticePoints(n=Ref(name='n')),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "GEOM", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "", "lemma_paths": [], "recipe_id": "", "seed_template_id": "geo_visible_lattice_v1", "ending_id": null, "olympiad_level": 5, "variant": "", "parent_id": "", "num_spawns": 0, "generation_time": 0.247, "created_at": "2026-02-08T03:56:58.060996Z", "verification": {"verified": true, "answer": 7351, "timestamp": "2026-02-08T03:56:58.308439Z"}, "problem_hash": "5d6674", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "answer": 7351, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 171, "completion_tokens": 5732}, "timestamp": "2026-02-09T14:52:36.604Z"}, {"id": 2, "model": "openai/gpt-oss-120b", "answer": 7263, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 167, "completion_tokens": 3793}, "timestamp": "2026-02-10T14:54:59.215Z"}, {"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 7351, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 103, "completion_tokens": 5806}, "timestamp": "2026-02-24T02:35:35.339Z"}, {"id": 8, "model": "mathstral", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 116, "completion_tokens": 498}, "timestamp": "2026-02-13T03:51:20.119Z"}, {"id": 10, "model": "qwen2-math:7b", "answer": 3246, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 111, "completion_tokens": 410}, "timestamp": "2026-02-27T18:47:18.018Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "answer": 108, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 162, "completion_tokens": 300}, "timestamp": "2026-02-11T21:16:47.819Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 11664, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 133, "completion_tokens": 548}, "timestamp": "2026-02-13T13:25:08.361Z"}, {"id": 36, "model": "qwen2.5:3b-instruct", "answer": 11881, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 114, "completion_tokens": 776}, "timestamp": "2026-04-18T12:09:50.640Z"}], "solution_status": 1, "irt_difficulty": {"lo": 1.97, "mid": 4.36, "hi": 6.69}}
{"id": "3a7a6b", "alias": "nt_count_divisible_and_v1_1125832087_1270", "problem": "Let $S$ be the set of all positive integers $t$ such that $11 \\leq t \\leq 8500$ and there exist positive integers $a$ and $b$ with $1 \\leq a \\leq 669$, $1 \\leq b \\leq 832$, and $t = 4a + 7b$. Let $N$ be the number of positive integers $n$ such that $1 \\leq n \\leq |S|$ and $n$ is divisible by both $6$ and $8$. Compute $N$.", "answer": 353, "graph": "graphs = [\n Graph(\n let={\n \"upper\": CountOverSet(set=SolutionsSet(var=Var(\"t\"), condition=Exists(var=Var(name='a'), condition=Exists(var=Var(name='b'), condition=And(Geq(left=Var(name='a'), right=Const(value=1)), Leq(left=Var(name='a'), right=Const(value=669)), Geq(left=Var(name='b'), right=Const(value=1)), Leq(left=Var(name='b'), right=Const(value=832)), Geq(left=Var(name='t'), right=Const(value=11)), Leq(left=Var(name='t'), right=Const(value=8500)), Eq(left=Var(name='t'), right=Sum(Mul(Const(value=4), Var(name='a')), Mul(Const(value=7), Var(name='b'))))))))),\n \"d1\": Const(6),\n \"d2\": Const(8),\n \"result\": CountOverSet(set=SolutionsSet(var=Var(\"n\"), condition=And(Geq(Var(\"n\"), Const(1)), Leq(Var(\"n\"), Ref(\"upper\")), Eq(Mod(value=Var(\"n\"), modulus=Ref(\"d1\")), Const(0)), Eq(Mod(value=Var(\"n\"), modulus=Ref(\"d2\")), Const(0))))),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "LIN_FORM", "lemma_paths": ["LIN_FORM"], "recipe_id": "7b2633", "seed_template_id": "nt_count_divisible_and_v1", "ending_id": null, "olympiad_level": 5, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["LIN_FORM"], "num_lemmas": 1, "generation_time": 0.669, "created_at": "2026-02-08T03:39:32.437516Z", "verification": {"verified": true, "answer": 353, "timestamp": "2026-02-08T03:39:33.106558Z"}, "problem_hash": "7c13a0", "license": "CC BY 4.0", "llm_solvers": [{"id": 2, "model": "openai/gpt-oss-120b", "answer": 353, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 235, "completion_tokens": 6311}, "timestamp": "2026-02-10T14:06:35.639Z"}, {"id": 8, "model": "mathstral", "answer": 354, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 183, "completion_tokens": 657}, "timestamp": "2026-02-12T21:39:16.212Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "answer": 64, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 226, "completion_tokens": 576}, "timestamp": "2026-02-11T21:05:47.904Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 354, "score": 1, "correct": {"strict": false, "boxed": false, "relaxed": true}, "usage": {"prompt_tokens": 201, "completion_tokens": 922}, "timestamp": "2026-02-13T08:15:31.900Z"}], "solution_status": 1, "lemma_applicability": {"LIN_FORM": "ok", "COUNT_FIB_DIVISIBLE": "no", "L3c": "no", "MAX_PRIME_BELOW": "no", "MOD_ADD": "no", "V3": "no"}, "irt_difficulty": {"lo": -3.49, "mid": 1.84, "hi": 7.55}}
{"id": "a36418", "alias": "sequence_lucas_compute_v1_809748730_82", "problem": "Let $m = 70245$. Let $n$ be the minimum value of $x + y$ over all ordered pairs of positive integers $(x, y)$ such that $xy = 5184$. Define $r = L_{19}$, the 19th Lucas number. Let $c$ be the sum of all solutions $x$ to the equation $x^2 - 25x + n = 0$. Find the remainder when $c - r$ is divided by $m$.", "answer": 60921, "graph": "graphs = [\n Graph(\n let={\n \"_m\": Const(70245),\n \"_n\": MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"x\"), Var(\"y\")]), condition=And(IsPositive(arg=Var(name='x')), IsPositive(arg=Var(name='y')), Eq(Mul(Var(\"x\"), Var(\"y\")), Const(5184)))), expr=Sum(Var(\"x\"), Var(\"y\")))),\n \"n\": Const(19),\n \"result\": Lucas(arg=Ref(name='n')),\n \"_c\": SumOverSet(set=SolutionsSet(var=Var(\"x\"), condition=Eq(Sum(Pow(Var(\"x\"), Const(2)), Mul(Const(-25), Var(\"x\")), Ref(\"_n\")), Const(0)))),\n \"Q\": Mod(value=Sub(Ref(\"_c\"), Ref(\"result\")), modulus=Ref(\"_m\")),\n },\n goal=Ref(\"Q\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "B3", "lemma_paths": ["B3/VIETA_SUM"], "recipe_id": "b9f41a", "seed_template_id": "sequence_lucas_compute_v1", "ending_id": "negation_mod", "olympiad_level": 5, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["B3", "VIETA_SUM"], "num_lemmas": 2, "generation_time": 0.003, "created_at": "2026-02-08T11:19:06.523742Z", "verification": {"verified": true, "answer": 60921, "timestamp": "2026-02-08T11:19:06.526993Z"}, "problem_hash": "a76134", "license": "CC BY 4.0", "llm_solvers": [{"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 60921, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 150, "completion_tokens": 1223}, "timestamp": "2026-02-14T11:42:05.548Z"}, {"id": 8, "model": "mathstral", "answer": 66089, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 203, "completion_tokens": 701}, "timestamp": "2026-02-21T16:22:18.533Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 214, "completion_tokens": 587}, "timestamp": "2026-02-12T03:47:36.947Z"}], "solution_status": 1, "lemma_applicability": {"B3": "ok", "VIETA_SUM": "ok_later", "K13": "no", "LTE_DIFF": "no", "LTE_DIFF_P2": "no", "MOD_ADD": "no", "V8_SUM": "no"}, "irt_difficulty": {"lo": -5.14, "mid": 0.32, "hi": 6.51}}
{"id": "bf1513", "alias": "comb_catalan_compute_v1_784195855_9519", "problem": "Let $n_0$ be the number of integers $t$ such that $13 \\le t \\le 59$ and there exist positive integers $a$ and $b$ with $1 \\le a \\le 4$, $1 \\le b \\le 8$, and $t = 6a + 4b + 3$. Let $n$ be the number of ordered pairs $(x_1, x_2)$ of positive odd integers such that $x_1 + x_2 = n_0$. Compute the $n$-th Catalan number.", "answer": 58786, "graph": "graphs = [\n Graph(\n let={\n \"_n\": CountOverSet(set=SolutionsSet(var=Var(\"t\"), condition=Exists(var=Var(name='a'), condition=Exists(var=Var(name='b'), condition=And(Geq(left=Var(name='a'), right=Const(value=1)), Leq(left=Var(name='a'), right=Const(value=4)), Geq(left=Var(name='b'), right=Const(value=1)), Leq(left=Var(name='b'), right=Const(value=8)), Geq(left=Var(name='t'), right=Const(value=13)), Leq(left=Var(name='t'), right=Const(value=59)), Eq(left=Var(name='t'), right=Sum(Mul(Const(value=6), Var(name='a')), Mul(Const(value=4), Var(name='b')), Const(value=3)))))))),\n \"n\": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"x1\"), Var(\"x2\")]), condition=And(IsPositive(arg=Var(name='x1')), IsPositive(arg=Var(name='x2')), IsOdd(arg=Var(name='x1')), IsOdd(arg=Var(name='x2')), Eq(Sum(Var(\"x1\"), Var(\"x2\")), Ref(\"_n\"))))),\n \"result\": Catalan(Ref(\"n\")),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "COMB", "secondary_domain": null, "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "LIN_FORM", "lemma_paths": ["LIN_FORM/COMB1"], "recipe_id": "268a62", "seed_template_id": "comb_catalan_compute_v1", "ending_id": null, "olympiad_level": 5, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["COMB1", "LIN_FORM"], "num_lemmas": 2, "generation_time": 0.003, "created_at": "2026-02-08T16:52:14.236328Z", "verification": {"verified": true, "answer": 58786, "timestamp": "2026-02-08T16:52:14.239764Z"}, "problem_hash": "82d955", "license": "CC BY 4.0", "llm_solvers": [{"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 58786, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 168, "completion_tokens": 2331}, "timestamp": "2026-02-17T13:54:34.651Z"}, {"id": 10, "model": "qwen2-math:7b", "answer": 14, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 177, "completion_tokens": 682}, "timestamp": "2026-03-01T00:16:27.847Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 226, "completion_tokens": 532}, "timestamp": "2026-02-12T21:15:34.935Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 1430, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 199, "completion_tokens": 1051}, "timestamp": "2026-02-25T21:48:59.832Z"}, {"id": 36, "model": "qwen2.5:3b-32k", "answer": 132, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 180, "completion_tokens": 1099}, "timestamp": "2026-04-21T01:30:15.712Z"}], "solution_status": 1, "lemma_applicability": {"LIN_FORM": "ok", "COMB1": "ok_later", "COUNT_CARTESIAN": "no", "COUNT_INTEGER_RANGE": "no", "COUNT_SUM_EQUALS": "no", "V7": "no", "V8": "no"}, "irt_difficulty": {"lo": -2.38, "mid": 1.74, "hi": 6.59}}
{"id": "f236d5_l", "alias": "geo_visible_lattice_v1_798873815_159", "problem": "Let $n = 200$. Define $r$ to be the number of visible lattice points $(x, y)$ with $1 \\le x, y \\le n$, where a point $(x, y)$ is visible if $\\gcd(x, y) = 1$. Compute the remainder when $\\sum_{k=1}^{r} \\tau(k)$ is divided by $96886$, where $\\tau(k)$ denotes the number of positive divisors of $k$.", "answer": 0, "graph": "", "domain": "GEOM", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "IDENTITY_POW_ZERO", "lemma_paths": ["IDENTITY_POW_ZERO"], "recipe_id": "bf46af", "seed_template_id": "geo_visible_lattice_v1", "ending_id": null, "olympiad_level": 6, "variant": "legacy_text", "parent_id": "f236d5", "num_spawns": 0, "lemma_set": ["IDENTITY_POW_ZERO"], "num_lemmas": 1, "generation_time": 0.936, "created_at": "2026-02-08T02:29:47.078122Z", "verification": {"verified": false, "answer": 57196, "error": "answer=0 != expected=57196", "timestamp": "2026-02-08T02:29:48.014581Z"}, "problem_hash": "91b13a", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 208, "completion_tokens": 32768}, "timestamp": "2026-02-23T14:11:54.309Z"}, {"id": 2, "model": "openai/gpt-oss-120b", "answer": 54455, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 208, "completion_tokens": 14119}, "timestamp": "2026-02-23T13:35:27.790Z"}, {"id": 3, "model": "Qwen/Qwen3-235B-A22B-Thinking-2507", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 161, "completion_tokens": 32768}, "timestamp": "2026-02-23T20:02:41.578Z"}, {"id": 4, "model": "NousResearch/Hermes-4-405B", "answer": 54482, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 155, "completion_tokens": 608}, "timestamp": "2026-02-08T20:49:08.442Z"}, {"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 0, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 143, "completion_tokens": 7979}, "timestamp": "2026-02-09T00:33:43.801Z"}, {"id": 8, "model": "mathstral", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 161, "completion_tokens": 760}, "timestamp": "2026-02-11T01:20:53.959Z"}, {"id": 10, "model": "qwen2-math:7b", "answer": 324, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 156, "completion_tokens": 505}, "timestamp": "2026-02-23T02:50:10.494Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 207, "completion_tokens": 566}, "timestamp": "2026-02-11T19:34:30.186Z"}, {"id": 15, "model": "Qwen/Qwen3-Coder-480B-A35B-Instruct", "answer": 29924, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 159, "completion_tokens": 2121}, "timestamp": "2026-02-12T00:00:11.604Z"}, {"id": 16, "model": "Qwen/Qwen3-Next-80B-A3B-Thinking", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 161, "completion_tokens": 32768}, "timestamp": "2026-02-23T22:17:49.010Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct-fast", "answer": 464, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 175, "completion_tokens": 730}, "timestamp": "2026-02-11T23:42:42.180Z"}, {"id": 29, "model": "Qwen/Qwen3-235B-A22B-Instruct-2507", "answer": 0, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 159, "completion_tokens": 15341}, "timestamp": "2026-02-23T14:48:13.327Z"}, {"id": 36, "model": "qwen2.5:3b-instruct", "answer": 13457, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 159, "completion_tokens": 956}, "timestamp": "2026-04-18T09:34:31.427Z"}], "solution_status": 0, "lemma_applicability": {"IDENTITY_POW_ZERO": "ok"}, "irt_difficulty": {"lo": 6.44, "mid": 8.24, "hi": 10.0}}
{"id": "b9b3b3", "alias": "alg_qf_psd_sum_v1_1218484723_7386", "problem": "Let $C = \\left|\\left\\{ (a_1, b_1) : 1 \\le a_1, b_1 \\le 15,\\, a_1 \\le b_1,\\, 32a_1^2 + 32b_1^2 - 64a_1b_1 = 1152 \\right\\}\\right|$. Compute the remainder when $$ \\sum_{a=1}^{202} \\sum_{b=1}^{202} \\left( C \\cdot b^2 + 30ab + 50a^2 \\right) $$ is divided by $61463$.", "answer": 51138, "graph": "graphs = [\n Graph(\n let={\n \"_n\": Const(1152),\n \"result\": Mod(value=SumOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"a\"), Var(\"b\")]), condition=And(Geq(Var(\"a\"), Const(1)), Leq(Var(\"a\"), Const(202)), Geq(Var(\"b\"), Const(1)), Leq(Var(\"b\"), Const(202)))), expr=Sum(Mul(CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"a1\"), Var(\"b1\")]), condition=And(Geq(Var(\"a1\"), Const(1)), Leq(Var(\"a1\"), Const(15)), Geq(Var(\"b1\"), Const(1)), Leq(Var(\"b1\"), Const(15)), Leq(Var(\"a1\"), Var(\"b1\")), Eq(Sum(Mul(Const(32), Pow(Var(\"b1\"), Const(2))), Mul(Const(32), Pow(Var(\"a1\"), Const(2))), Mul(Const(-64), Var(\"a1\"), Var(\"b1\"))), Ref(\"_n\"))))), Pow(Var(\"b\"), Const(2))), Mul(Const(30), Var(\"a\"), Var(\"b\")), Mul(Const(50), Pow(Var(\"a\"), Const(2)))))), modulus=Const(61463)),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "ALG", "secondary_domain": null, "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "QF_PSD_ORBIT", "lemma_paths": ["QF_PSD_ORBIT"], "recipe_id": "1d37f3", "seed_template_id": "alg_qf_psd_sum_v1", "ending_id": null, "olympiad_level": 6, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["QF_PSD_ORBIT"], "num_lemmas": 1, "generation_time": 0.075, "created_at": "2026-02-25T08:48:06.272148Z", "verification": {"verified": true, "answer": 51138, "timestamp": "2026-02-25T08:48:06.346752Z"}, "problem_hash": "1421c4", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "answer": 51138, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 246, "completion_tokens": 7942}, "timestamp": "2026-03-30T04:17:38.838Z"}, {"id": 2, "model": "openai/gpt-oss-120b", "answer": 51138, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 246, "completion_tokens": 5039}, "timestamp": "2026-05-03T11:52:22.193Z"}, {"id": 16, "model": "Qwen/Qwen3-Next-80B-A3B-Thinking", "answer": 51138, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 210, "completion_tokens": 10947}, "timestamp": "2026-03-30T21:30:36.192Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 0, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 214, "completion_tokens": 1174}, "timestamp": "2026-04-20T02:51:55.544Z"}, {"id": 36, "model": "qwen2.5:3b-32k", "answer": 81, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 208, "completion_tokens": 1458}, "timestamp": "2026-04-21T23:22:06.121Z"}], "solution_status": 1, "lemma_applicability": {"QF_PSD_ORBIT": "ok", "POLY_PADIC_VAL_CONST": "no"}, "irt_difficulty": {"lo": -3.33, "mid": 1.03, "hi": 5.26}}
{"id": "041063", "alias": "alg_poly_orbit_count_v1_1218484723_40", "problem": "For a non-negative integer $a$, define the sequence:\n\\[\nN = (2a^4 - 2a^3 - 5a^2 - 2a + 1) \\bmod 61,\\quad M = (2N^4 - 2N^3 - 5N^2 - 2N + 1) \\bmod 61,\n\\]\n\\[\nR = (2M^4 - 2M^3 - 5M^2 - 2M + 1) \\bmod 61,\\quad S = (2R^4 - 2R^3 - 5R^2 - 2R + 1) \\bmod 61,\n\\]\n\\[\nT = (2S^4 - 2S^3 - 5S^2 - 2S + 1) \\bmod 61.\n\\]\nLet $Q$ be the number of integers $a$ with $0 \\le a \\le 111751$ such that $T = a$ but $N \\ne a$, $M \\ne a$, $R \\ne a$, and $S \\ne a$. Find $Q$.", "answer": 9160, "graph": "graphs = [\n Graph(\n let={\n \"p1\": Mod(value=Sum(Mul(Const(2), Pow(Var(\"a\"), Const(4))), Mul(Const(-2), Pow(Var(\"a\"), Const(3))), Mul(Const(-5), Pow(Var(\"a\"), Const(2))), Mul(Const(-2), Var(\"a\")), Const(1)), modulus=Const(61)),\n \"p2\": Mod(value=Sum(Mul(Const(2), Pow(Ref(\"p1\"), Const(4))), Mul(Const(-2), Pow(Ref(\"p1\"), Const(3))), Mul(Const(-5), Pow(Ref(\"p1\"), Const(2))), Mul(Const(-2), Ref(\"p1\")), Const(1)), modulus=Const(61)),\n \"p3\": Mod(value=Sum(Mul(Const(2), Pow(Ref(\"p2\"), Const(4))), Mul(Const(-2), Pow(Ref(\"p2\"), Const(3))), Mul(Const(-5), Pow(Ref(\"p2\"), Const(2))), Mul(Const(-2), Ref(\"p2\")), Const(1)), modulus=Const(61)),\n \"p4\": Mod(value=Sum(Mul(Const(2), Pow(Ref(\"p3\"), Const(4))), Mul(Const(-2), Pow(Ref(\"p3\"), Const(3))), Mul(Const(-5), Pow(Ref(\"p3\"), Const(2))), Mul(Const(-2), Ref(\"p3\")), Const(1)), modulus=Const(61)),\n \"p5\": Mod(value=Sum(Mul(Const(2), Pow(Ref(\"p4\"), Const(4))), Mul(Const(-2), Pow(Ref(\"p4\"), Const(3))), Mul(Const(-5), Pow(Ref(\"p4\"), Const(2))), Mul(Const(-2), Ref(\"p4\")), Const(1)), modulus=Const(61)),\n \"result\": CountOverSet(set=SolutionsSet(var=Var(\"a\"), condition=And(Geq(Var(\"a\"), Const(0)), Leq(Var(\"a\"), Const(111751)), Eq(Ref(\"p5\"), Var(\"a\")), Neq(Ref(\"p1\"), Var(\"a\")), Neq(Ref(\"p2\"), Var(\"a\")), Neq(Ref(\"p3\"), Var(\"a\")), Neq(Ref(\"p4\"), Var(\"a\"))))),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "ALG", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "POLY_ORBIT_COUNT", "lemma_paths": ["POLY_ORBIT_COUNT"], "recipe_id": "4ad965", "seed_template_id": "alg_poly_orbit_count_v1", "ending_id": null, "olympiad_level": 6, "variant": "", "parent_id": "", "lemma_set": ["POLY_ORBIT_COUNT"], "num_lemmas": 1, "generation_time": 0.113, "created_at": "2026-02-25T01:44:35.135876Z", "verification": {"verified": true, "answer": 9160, "timestamp": "2026-02-25T01:44:35.248619Z"}, "problem_hash": "844979", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 362, "completion_tokens": 32768}, "timestamp": "2026-03-10T08:17:19.477Z"}, {"id": 2, "model": "openai/gpt-oss-120b", "answer": 9160, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 362, "completion_tokens": 7370}, "timestamp": "2026-04-18T14:20:12.888Z"}, {"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 109920, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 299, "completion_tokens": 3995}, "timestamp": "2026-04-18T23:33:35.941Z"}, {"id": 10, "model": "qwen2-math:7b", "answer": 56, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 314, "completion_tokens": 802}, "timestamp": "2026-03-10T08:06:26.600Z"}, {"id": 16, "model": "Qwen/Qwen3-Next-80B-A3B-Thinking", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 319, "completion_tokens": 32768}, "timestamp": "2026-03-28T21:51:31.205Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 30, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 329, "completion_tokens": 1131}, "timestamp": "2026-04-18T23:04:11.573Z"}, {"id": 36, "model": "qwen2.5:3b-32k", "answer": 0, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 317, "completion_tokens": 821}, "timestamp": "2026-04-21T06:34:54.244Z"}, {"id": 38, "model": "google/gemma-3-27b-it", "answer": 111751, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 315, "completion_tokens": 1910}, "timestamp": "2026-04-18T23:23:32.418Z"}], "solution_status": 1, "lemma_applicability": {"POLY_ORBIT_COUNT": "ok", "POLY_PADIC_VAL_CONST": "no"}, "irt_difficulty": {"lo": 3.86, "mid": 5.72, "hi": 7.83}}
{"id": "1c47f1", "alias": "antilemma_k3_v1_784195855_2193", "problem": "Let $S$ be the set of all positive integers $x$ such that $x^2 - 3236x + 313600 = 0$. Let $n$ be the sum of all elements in $S$. Compute the value of $$\nx = \\sum_{d \\mid n} \\phi(d),$$ where $\\phi$ denotes Euler's totient function.", "answer": 3236, "graph": "graphs = [\n Graph(\n let={\n \"_n\": Const(2),\n \"x\": SumOverDivisors(n=SumOverSet(set=SolutionsSet(var=Var(name='x'), condition=Eq(left=Sum(Pow(base=Var(name='x'), exp=Ref(name='_n')), Mul(Const(value=-3236), Var(name='x')), Const(value=313600)), right=Const(value=0)))), var='d', expr=EulerPhi(n=Var(name='d'))),\n },\n goal=Ref(\"x\"),\n )\n]", "domain": "NT", "secondary_domain": "COMB", "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "VIETA_SUM", "lemma_paths": ["VIETA_SUM/K3", "K3"], "recipe_id": "78a626", "seed_template_id": "antilemma_k3_v1", "ending_id": null, "olympiad_level": 6, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["K3", "VIETA_SUM"], "num_lemmas": 2, "generation_time": 0.001, "created_at": "2026-02-08T05:34:33.436879Z", "verification": {"verified": true, "answer": 3236, "timestamp": "2026-02-08T05:34:33.437510Z"}, "problem_hash": "da4b3f", "license": "CC BY 4.0", "llm_solvers": [{"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 3236, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 124, "completion_tokens": 947}, "timestamp": "2026-02-12T10:44:40.015Z"}, {"id": 8, "model": "mathstral", "answer": 5130, "score": 1, "correct": {"strict": false, "boxed": false, "relaxed": true}, "usage": {"prompt_tokens": 144, "completion_tokens": 775}, "timestamp": "2026-02-14T19:55:01.728Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "answer": 3243, "score": 1, "correct": {"strict": false, "boxed": false, "relaxed": true}, "usage": {"prompt_tokens": 190, "completion_tokens": 705}, "timestamp": "2026-02-11T22:48:35.653Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 3236, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 156, "completion_tokens": 671}, "timestamp": "2026-02-14T20:37:53.224Z"}], "solution_status": 1, "lemma_applicability": {"K3": "ok", "VIETA_SUM": "ok", "DS2": "no", "K17": "no", "L3c": "no", "MAX_VAL": "no", "V1": "no"}, "irt_difficulty": {"lo": -5.55, "mid": -3.01, "hi": 0.32}}
{"id": "3be28a", "alias": "modular_count_residue_v1_1431428450_739", "problem": "Let $r$ be the value of $\\frac{7}{49} \\sum_{k=1}^7 \\sum_{j=1}^7 \\phi(k) \\left\\lfloor \\frac{7}{k} \\right\\rfloor$, where $\\phi$ denotes Euler's totient function. Let $N$ be the number of positive integers $n$ such that $n \\le 84100$ and $n \\equiv r \\pmod{30}$. Compute the remainder when $45816 \\cdot N$ is divided by 91801.", "answer": 84450, "graph": "graphs = [\n Graph(\n let={\n \"upper\": Const(84100),\n \"m\": Const(30),\n \"r\": Div(Mul(Const(7), SumOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"k\"), Var(\"_j\")]), condition=Const(1), domain=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(7)), right=IntegerRange(start=Const(1), end=Const(7)))), expr=Mul(EulerPhi(n=Var(\"k\")), Floor(Div(Const(7), Var(\"k\"))))))), Const(49)),\n \"result\": CountOverSet(set=SolutionsSet(var=Var(\"n\"), condition=And(Geq(Var(\"n\"), Const(1)), Leq(Var(\"n\"), Ref(\"upper\")), Eq(Mod(value=Var(\"n\"), modulus=Ref(\"m\")), Ref(\"r\"))))),\n \"_c\": Const(45816),\n \"Q\": Mod(value=Mul(Ref(\"_c\"), Ref(\"result\")), modulus=Const(91801)),\n },\n goal=Ref(\"Q\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "SUM_INDEPENDENT", "lemma_paths": ["SUM_INDEPENDENT", "K2"], "recipe_id": "d64c9f", "seed_template_id": "modular_count_residue_v1", "ending_id": null, "olympiad_level": 6, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["K2", "SUM_INDEPENDENT"], "num_lemmas": 2, "generation_time": 4.51, "created_at": "2026-02-08T13:39:22.636562Z", "verification": {"verified": true, "answer": 84450, "timestamp": "2026-02-08T13:39:27.146525Z"}, "problem_hash": "397cdc", "license": "CC BY 4.0", "llm_solvers": [{"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 84450, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 161, "completion_tokens": 2057}, "timestamp": "2026-02-15T19:05:17.822Z"}, {"id": 8, "model": "mathstral", "answer": 6557, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 224, "completion_tokens": 1171}, "timestamp": "2026-02-25T10:18:56.938Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 231, "completion_tokens": 501}, "timestamp": "2026-02-12T07:12:13.305Z"}], "solution_status": 1, "lemma_applicability": {"K2": "ok", "SUM_INDEPENDENT": "ok", "COUNT_DIVISOR_COUNT": "no", "COUNT_FIB_DIVISIBLE": "no", "DS2": "no", "MOD_SUB": "no", "V3": "no"}, "irt_difficulty": {"lo": -5.49, "mid": 0.22, "hi": 6.51}}
{"id": "94a701", "alias": "sequence_fibonacci_compute_v1_151522320_67", "problem": "Let $S$ be the set of all integers $n$ such that $1 \\leq n \\leq 110$ and $n \\equiv \\left\\lfloor \\frac{n}{2} \\right\\rfloor \\pmod{5}$. Let $k$ be the number of elements in $S$. Compute the $k$-th Fibonacci number, where the Fibonacci sequence is defined by $F_1 = 1$, $F_2 = 1$, and $F_m = F_{m-1} + F_{m-2}$ for $m \\geq 3$. Determine the value of this Fibonacci number.", "answer": 17711, "graph": "graphs = [\n Graph(\n let={\n \"_n\": Const(110),\n \"n\": CountOverSet(set=SolutionsSet(var=Var(\"n\"), condition=And(Geq(Var(\"n\"), Const(1)), Leq(Var(\"n\"), Ref(\"_n\")), Congruent(a=Var(name='n'), b=Floor(arg=Div(left=Var(name='n'), right=Const(value=2))), modulus=Const(value=5))))),\n \"result\": Fibonacci(arg=Ref(name='n')),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "L3C", "lemma_paths": ["L3C"], "recipe_id": "73f8b0", "seed_template_id": "sequence_fibonacci_compute_v1", "ending_id": null, "olympiad_level": 6, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["L3C"], "num_lemmas": 1, "generation_time": 0.001, "created_at": "2026-02-08T02:56:30.778343Z", "verification": {"verified": true, "answer": 17711, "timestamp": "2026-02-08T02:56:30.779561Z"}, "problem_hash": "9d1796", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "answer": 17711, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 248, "completion_tokens": 1503}, "timestamp": "2026-02-10T11:57:51.334Z"}, {"id": 2, "model": "openai/gpt-oss-120b", "answer": 17711, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 252, "completion_tokens": 1123}, "timestamp": "2026-02-08T22:25:47.545Z"}, {"id": 4, "model": "NousResearch/Hermes-4-405B", "answer": 17711, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 193, "completion_tokens": 1076}, "timestamp": "2026-02-10T16:31:18.700Z"}, {"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 17711, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 176, "completion_tokens": 939}, "timestamp": "2026-02-10T20:04:00.118Z"}, {"id": 8, "model": "mathstral", "answer": 1134903170, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 201, "completion_tokens": 656}, "timestamp": "2026-02-11T08:12:07.400Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "answer": 55, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 236, "completion_tokens": 961}, "timestamp": "2026-02-11T19:57:16.867Z"}, {"id": 15, "model": "Qwen/Qwen3-Coder-480B-A35B-Instruct", "answer": 17711, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 194, "completion_tokens": 923}, "timestamp": "2026-02-12T02:19:30.691Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct-fast", "answer": 354224848179261915075, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 213, "completion_tokens": 999}, "timestamp": "2026-02-12T00:35:33.808Z"}, {"id": 29, "model": "Qwen/Qwen3-235B-A22B-Instruct-2507", "answer": 17711, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 194, "completion_tokens": 1454}, "timestamp": "2026-02-23T19:56:15.581Z"}], "solution_status": 1, "lemma_applicability": {"L3C": "ok", "COUNT_COPRIME_GRID": "no", "K14": "no", "L3b": "no", "MOD_ADD": "no", "V7": "no"}, "irt_difficulty": {"lo": -2.25, "mid": 0.01, "hi": 1.86}}
{"id": "ada62f", "alias": "comb_catalan_compute_v1_601307018_6638", "problem": "Let $C_n$ denote the $n$-th Catalan number, and let $n = \\sum_{k=1}^{4} \\varphi(k) \\cdot \\left\\lfloor \\frac{4}{k} \\right\\rfloor$. Let $T = C_n$. Define $u = \\sum_{k=0}^{0} (-1)^k \\binom{0}{k}$, $v = \\sum_{k=0}^{8} (-1)^k \\binom{8}{k}$, $a = (4 + v) \\cdot u$, $K = a + 4$, and $s = \\sum_{k=0}^{K} (-1)^k \\binom{K}{k}$. Find the remainder when $14584 \\cdot T$ is divided by $61991 + s$.", "answer": 26423, "graph": "graphs = [\n Graph(\n let={\n \"_n\": Const(4),\n \"n3\": Const(8),\n \"v\": Summation(var=\"k\", start=Const(0), end=Ref(\"n3\"), expr=Mul(Pow(Const(-1), Var(\"k\")), Binom(n=Ref(\"n3\"), k=Var(\"k\")))),\n \"n2\": Const(0),\n \"u\": Summation(var=\"k1\", start=Const(0), end=Ref(\"n2\"), expr=Mul(Pow(Const(-1), Var(\"k1\")), Binom(n=Ref(\"n2\"), k=Var(\"k1\")))),\n \"a\": Mul(Sum(Const(4), Ref(\"v\")), Ref(\"u\")),\n \"b\": Const(4),\n \"n1\": Sum(Ref(\"a\"), Ref(\"b\")),\n \"s\": Summation(var=\"k2\", start=Const(0), end=Ref(\"n1\"), expr=Mul(Pow(Const(-1), Var(\"k2\")), Binom(n=Ref(\"n1\"), k=Var(\"k2\")))),\n \"n\": Summation(var=\"k3\", start=Const(1), end=Ref(\"_n\"), expr=Mul(EulerPhi(n=Var(\"k3\")), Floor(Div(Const(4), Var(\"k3\"))))),\n \"result\": Catalan(Ref(\"n\")),\n \"_c\": Const(14584),\n \"Q\": Mod(value=Mul(Ref(\"_c\"), Ref(\"result\")), modulus=Sum(Const(61991), Ref(\"s\"))),\n },\n goal=Ref(\"Q\"),\n )\n]", "domain": "COMB", "secondary_domain": "NT", "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "BINOMIAL_ALTERNATING", "lemma_paths": ["BINOMIAL_ALTERNATING", "K2"], "recipe_id": "1ce58e", "seed_template_id": "comb_catalan_compute_v1", "ending_id": null, "olympiad_level": 6, "variant": "", "parent_id": "", "num_spawns": 3, "lemma_set": ["BINOMIAL_ALTERNATING", "K2"], "num_lemmas": 2, "generation_time": 0.005, "created_at": "2026-03-10T07:17:38.339876Z", "verification": {"verified": true, "answer": 26423, "timestamp": "2026-03-10T07:17:38.345128Z"}, "problem_hash": "ed09db", "license": "CC BY 4.0", "llm_solvers": [{"id": 2, "model": "openai/gpt-oss-120b", "answer": 26423, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 292, "completion_tokens": 1041}, "timestamp": "2026-04-19T04:55:13.786Z"}, {"id": 36, "model": "qwen2.5:3b-32k", "answer": 5912, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 244, "completion_tokens": 1038}, "timestamp": "2026-04-22T16:53:11.318Z"}], "solution_status": 1, "lemma_applicability": {"BINOMIAL_ALTERNATING": "ok", "K2": "ok", "C2": "no", "COUNT_CARTESIAN": "no", "COUNT_COPRIME_GRID": "no", "COUNT_INTEGER_RANGE": "no", "V8_SUM": "no"}, "irt_difficulty": {"lo": -5.34, "mid": 1.1, "hi": 7.53}}
{"id": "d520de", "alias": "sequence_count_fib_divisible_v1_865884756_5881", "problem": "Let $n$ be a positive integer such that $n \\leq 75076$ and $n$ divides $75076$. Define $s(n)$ to be the sum of the positive divisors $a$ and $b$ such that $a \\cdot b = 75076$. Let $s_{\\min}$ be the minimum possible value of $s(n)$ over all such $n$. Now consider all integers $t$ for which there exist integers $a$ and $b$ satisfying:\n\n- $1 \\leq a \\leq 9$,\n- $1 \\leq b \\leq 2$,\n- $23 \\leq t \\leq 46$,\n- $t = 2a + 7b + 14$.\n\nLet $d$ be the number of such integers $t$. Determine the number of positive integers $n$ such that $1 \\leq n \\leq s_{\\min}$ and $d$ divides the $n$-th Fibonacci number, where the Fibonacci sequence is defined by $F_1 = 1$, $F_2 = 1$, and $F_k = F_{k-1} + F_{k-2}$ for $k \\geq 3$.", "answer": 45, "graph": "graphs = [\n Graph(\n let={\n \"_n\": Const(75076),\n \"upper\": MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"x\"), Var(\"y\")]), condition=And(IsPositive(arg=Var(name='x')), IsPositive(arg=Var(name='y')), Eq(Mul(Var(\"x\"), Var(\"y\")), Ref(\"_n\")))), expr=Sum(Var(\"x\"), Var(\"y\")))),\n \"d\": CountOverSet(set=SolutionsSet(var=Var(\"t\"), condition=Exists(var=Var(name='a'), condition=Exists(var=Var(name='b'), condition=And(Geq(left=Var(name='a'), right=Const(value=1)), Leq(left=Var(name='a'), right=Const(value=9)), Geq(left=Var(name='b'), right=Const(value=1)), Leq(left=Var(name='b'), right=Const(value=2)), Geq(left=Var(name='t'), right=Const(value=23)), Leq(left=Var(name='t'), right=Const(value=46)), Eq(left=Var(name='t'), right=Sum(Mul(Const(value=2), Var(name='a')), Mul(Const(value=7), Var(name='b')), Const(value=14)))))))),\n \"result\": CountOverSet(set=SolutionsSet(var=Var(\"n\"), condition=And(Geq(Var(\"n\"), Const(1)), Leq(Var(\"n\"), Ref(\"upper\")), Divides(divisor=Ref(\"d\"), dividend=Fibonacci(arg=Var(name='n')))))),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "ONE_PHI_2", "lemma_paths": ["LIN_FORM", "B3"], "recipe_id": "688dbe", "seed_template_id": "sequence_count_fib_divisible_v1", "ending_id": null, "olympiad_level": 6, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["B3", "LIN_FORM", "ONE_PHI_2"], "num_lemmas": 3, "generation_time": 0.052, "created_at": "2026-02-08T18:50:54.655842Z", "verification": {"verified": true, "answer": 45, "timestamp": "2026-02-08T18:50:54.707811Z"}, "problem_hash": "dc18cc", "license": "CC BY 4.0", "llm_solvers": [{"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 45, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 297, "completion_tokens": 2959}, "timestamp": "2026-02-18T19:44:54.263Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 370, "completion_tokens": 457}, "timestamp": "2026-02-13T00:33:50.083Z"}], "solution_status": 1, "lemma_applicability": {"B3": "ok", "LIN_FORM": "ok", "COUNT_FIB_DIVISIBLE": "no", "COUNT_PRIMES": "no", "MAX_VAL": "no", "V8": "no", "VAL_SUM_EQ": "no"}, "irt_difficulty": {"lo": -7.08, "mid": -0.29, "hi": 6.49}}
{"id": "161cb3", "alias": "nt_sum_over_divisible_v1_898971024_2499", "problem": "Let $a_k = (-1)^k \\binom{9}{k}$ for $k = 0, 1, \\dots, 9$, and let $s = \\sum_{k=0}^{9} a_k$. Let $U$ be the set of all integers $n$ such that $1 \\leq n \\leq 19321$ and $n \\equiv s \\pmod{145}$. Let $r$ be the sum of all elements in $U$. Let $V$ be the set of all ordered pairs $(x, y)$ of positive integers such that $x + y = 160$, and let $c$ be the maximum value of $xy$ over all such pairs. Find the value of $(r^2 + 11r + c) \\bmod{53517}$.", "answer": 44641, "graph": "graphs = [\n Graph(\n let={\n \"upper\": Const(19321),\n \"divisor\": Const(145),\n \"result\": SumOverSet(set=SolutionsSet(var=Var(\"n\"), condition=And(Geq(Var(\"n\"), Const(1)), Leq(Var(\"n\"), Ref(\"upper\")), Eq(Mod(value=Var(\"n\"), modulus=Ref(\"divisor\")), Summation(var=\"k\", start=Const(0), end=Const(9), expr=Mul(Pow(Const(-1), Var(\"k\")), Binom(n=Const(9), k=Var(\"k\")))))))),\n \"_c\": MaxOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"x\"), Var(\"y\")]), condition=And(IsPositive(arg=Var(name='x')), IsPositive(arg=Var(name='y')), Eq(Sum(Var(\"x\"), Var(\"y\")), Const(160)))), expr=Mul(Var(\"x\"), Var(\"y\")))),\n \"Q\": Mod(value=Sum(Pow(Ref(\"result\"), Const(2)), Mul(Const(11), Ref(\"result\")), Ref(\"_c\")), modulus=Const(53517)),\n },\n goal=Ref(\"Q\"),\n )\n]", "domain": "COMB", "secondary_domain": null, "goal": "SUM", "evaluator_id": "sympy", "root_lemma": "B1", "lemma_paths": ["B1", "BINOMIAL_ALTERNATING"], "recipe_id": "4bb7d3", "seed_template_id": "nt_sum_over_divisible_v1", "ending_id": "quadratic_mod", "olympiad_level": 6, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["B1", "BINOMIAL_ALTERNATING"], "num_lemmas": 2, "generation_time": 0.923, "created_at": "2026-02-08T16:47:09.152563Z", "verification": {"verified": true, "answer": 44641, "timestamp": "2026-02-08T16:47:10.075328Z"}, "problem_hash": "f048ea", "license": "CC BY 4.0", "llm_solvers": [{"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 44641, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 217, "completion_tokens": 1565}, "timestamp": "2026-02-17T12:56:22.902Z"}, {"id": 10, "model": "qwen2-math:7b", "answer": 117, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 240, "completion_tokens": 1001}, "timestamp": "2026-03-01T00:06:27.056Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 285, "completion_tokens": 496}, "timestamp": "2026-02-12T21:09:50.481Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 47189, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 253, "completion_tokens": 855}, "timestamp": "2026-02-25T21:42:58.249Z"}, {"id": 36, "model": "qwen2.5:3b-32k", "answer": 1469, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 243, "completion_tokens": 1628}, "timestamp": "2026-04-21T01:14:32.608Z"}], "solution_status": 1, "lemma_applicability": {"B1": "ok", "BINOMIAL_ALTERNATING": "ok", "C2": "no", "COUNT_COPRIME_GRID": "no", "COUNT_INTEGER_RANGE": "no", "COUNT_SUM_EQUALS": "no", "V8": "no"}, "irt_difficulty": {"lo": -2.38, "mid": 1.74, "hi": 6.59}}
{"id": "d4f8a4", "alias": "antilemma_k2_v1_971394319_299", "problem": "Let $S$ be the set of all real solutions $x$ to the equation $x^2 - 246x - 17271 = 0$. Let $m$ be the sum of all elements in $S$. Define\n$$\nx = \\sum_{k=1}^{m} \\phi(k) \\cdot \\left\\lfloor \\frac{246}{k} \\right\\rfloor,\n$$\nwhere $\\phi(k)$ is Euler's totient function. Let $Q$ be the remainder when $80050 \\cdot x$ is divided by $92349$. Find the value of $Q$.", "answer": 80484, "graph": "graphs = [\n Graph(\n let={\n \"x\": Summation(var=\"k\", start=Const(1), end=SumOverSet(set=SolutionsSet(var=Var(\"x\"), condition=Eq(Sum(Pow(Var(\"x\"), Const(2)), Mul(Const(-246), Var(\"x\")), Const(-17271)), Const(0)))), expr=Mul(EulerPhi(n=Var(\"k\")), Floor(Div(Const(246), Var(\"k\"))))),\n \"_c\": Const(80050),\n \"Q\": Mod(value=Mul(Ref(\"_c\"), Ref(\"x\")), modulus=Const(92349)),\n },\n goal=Ref(\"Q\"),\n )\n]", "domain": "NT", "secondary_domain": "COMB", "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "VIETA_SUM", "lemma_paths": ["VIETA_SUM/K2", "K2"], "recipe_id": "6be084", "seed_template_id": "antilemma_k2_v1", "ending_id": null, "olympiad_level": 6, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["K2", "VIETA_SUM"], "num_lemmas": 2, "generation_time": 0.002, "created_at": "2026-02-08T12:56:57.796565Z", "verification": {"verified": true, "answer": 80484, "timestamp": "2026-02-08T12:56:57.798170Z"}, "problem_hash": "a5e9c3", "license": "CC BY 4.0", "llm_solvers": [{"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 80484, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 171, "completion_tokens": 1845}, "timestamp": "2026-02-15T08:08:14.918Z"}, {"id": 8, "model": "mathstral", "answer": 246, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 238, "completion_tokens": 276}, "timestamp": "2026-02-22T16:55:07.560Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 247, "completion_tokens": 472}, "timestamp": "2026-02-12T05:59:34.797Z"}], "solution_status": 1, "lemma_applicability": {"K2": "ok", "VIETA_SUM": "ok", "DS2": "no", "MAX_VAL": "no", "V3": "no", "V7": "no", "V8_SUM": "no"}, "irt_difficulty": {"lo": -5.14, "mid": 0.32, "hi": 6.51}}
{"id": "6dae95_l", "alias": "comb_catalan_compute_v1_784195855_17", "problem": "Let $N$ be the number of ordered triples $(x_1,x_2,x_3)$ of positive integers such that each of $x_1,x_2,x_3$ is odd and\n$$x_1+x_2+x_3=7.$$\nLet $K=\\binom{6}{N}$ and let\n$$F=K!.$$\nLet $C_{11}$ be the $11$th Catalan number, and let\n$$S=\\sum_{n=F}^{\\lvert C_{11}\\rvert} d(n),$$\nwhere $d(n)$ denotes the number of positive divisors of $n$.\nLet $Q$ be the remainder when $S$ is divided by $98639$.\n\nFind the value of $Q$.", "answer": 0, "graph": "", "domain": "COMB", "secondary_domain": null, "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "COMB1", "lemma_paths": ["COMB1/ZERO_BINOM_N/ONE_FACTORIAL_0"], "recipe_id": "b96107", "seed_template_id": "comb_catalan_compute_v1", "ending_id": "sum_divisor_count", "olympiad_level": 7, "variant": "legacy_text", "parent_id": "6dae95", "num_spawns": 0, "lemma_set": ["COMB1", "ONE_FACTORIAL_0", "ZERO_BINOM_N"], "num_lemmas": 3, "generation_time": 0.003, "created_at": "2026-02-08T02:54:14.193493Z", "verification": {"verified": false, "answer": 62818, "error": "answer=0 != expected=62818", "timestamp": "2026-02-08T02:54:14.196332Z"}, "problem_hash": "c5d351", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 274, "completion_tokens": 32768}, "timestamp": "2026-02-23T19:52:51.246Z"}, {"id": 2, "model": "openai/gpt-oss-120b", "answer": 62776, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 274, "completion_tokens": 13453}, "timestamp": "2026-02-23T14:56:36.484Z"}, {"id": 3, "model": "Qwen/Qwen3-235B-A22B-Thinking-2507", "answer": 62816, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 227, "completion_tokens": 29353}, "timestamp": "2026-02-24T00:29:02.162Z"}, {"id": 4, "model": "NousResearch/Hermes-4-405B", "answer": 1999, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 220, "completion_tokens": 807}, "timestamp": "2026-02-10T16:11:03.688Z"}, {"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 58786, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 203, "completion_tokens": 3906}, "timestamp": "2026-02-10T19:11:06.831Z"}, {"id": 8, "model": "mathstral", "answer": 0, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 236, "completion_tokens": 404}, "timestamp": "2026-02-11T07:41:12.613Z"}, {"id": 10, "model": "qwen2-math:7b", "answer": 35, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 222, "completion_tokens": 1026}, "timestamp": "2026-02-27T15:08:58.986Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 277, "completion_tokens": 456}, "timestamp": "2026-02-11T19:55:40.178Z"}, {"id": 15, "model": "Qwen/Qwen3-Coder-480B-A35B-Instruct", "answer": 0, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 225, "completion_tokens": 1235}, "timestamp": "2026-02-12T02:10:10.577Z"}, {"id": 16, "model": "Qwen/Qwen3-Next-80B-A3B-Thinking", "answer": 62814, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 227, "completion_tokens": 30616}, "timestamp": "2026-02-23T23:12:31.876Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct-fast", "answer": 0, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 240, "completion_tokens": 914}, "timestamp": "2026-02-12T00:31:59.529Z"}, {"id": 29, "model": "Qwen/Qwen3-235B-A22B-Instruct-2507", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 225, "completion_tokens": 32768}, "timestamp": "2026-02-23T19:39:33.579Z"}, {"id": 36, "model": "qwen2.5:3b-instruct", "answer": 1, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 225, "completion_tokens": 1130}, "timestamp": "2026-04-18T10:02:25.227Z"}], "solution_status": 0, "lemma_applicability": {"COMB1": "ok", "ONE_FACTORIAL_0": "ok_later", "ZERO_BINOM_N": "ok_later", "COUNT_CARTESIAN": "no", "COUNT_COPRIME_GRID": "no", "COUNT_INTEGER_RANGE": "no", "COUNT_SUM_EQUALS": "no", "V7": "no"}, "irt_difficulty": {"lo": 6.44, "mid": 8.24, "hi": 10.0}}
{"id": "d7899e", "alias": "geo_count_lattice_triangle_v1_784195855_6908", "problem": "Let $A$ be twice the area of a triangle with vertices at $(0, 0)$, $(222, 0)$, and $(0, 111)$, adjusted by subtracting $49$ times the number of ordered pairs $(i, j)$ of positive integers such that $i + j = 112$, $1 \\leq i \\leq 111$, and $1 \\leq j \\leq 112$. Let $B$ be the sum of the greatest common divisors of the following pairs: $\\left( \\left| \\text{number of lattice points in } [1,6] \\times [1,37] \\right|, 49 \\right)$, $\\left( \\left| 111 - m \\right|, \\left| 111 - 49 \\right| \\right)$ where $m$ is the minimum value of $x+y$ over all pairs of positive integers $(x, y)$ such that $xy = 12321$, and $\\left( 111, 111 \\right)$. Compute $\\frac{A + 2 - B}{2}$.", "answer": 9546, "graph": "graphs = [\n Graph(\n let={\n \"_c\": Const(49),\n \"_m\": Const(49),\n \"_n\": Const(111),\n \"area_2x\": Abs(arg=Sum(Mul(Const(value=222), Const(value=111)), Mul(CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var(name='i'), Var(name='j')]), condition=Eq(left=Sum(Var(name='i'), Var(name='j')), right=Const(value=112)), domain=CartesianProduct(left=IntegerRange(start=Const(value=1), end=Const(value=111)), right=IntegerRange(start=Const(value=1), end=Const(value=112))))), Sub(left=Const(value=0), right=Ref(name='_c'))))),\n \"boundary\": Sum(GCD(a=Abs(arg=CountOverSet(set=CartesianProduct(left=IntegerRange(start=Const(value=1), end=Const(value=6)), right=IntegerRange(start=Const(value=1), end=Const(value=37))))), b=Abs(arg=Const(value=49))), GCD(a=Abs(arg=Sub(left=Const(value=111), right=MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var(name='x'), Var(name='y')]), condition=And(IsPositive(arg=Var(name='x')), IsPositive(arg=Var(name='y')), Eq(left=Mul(Var(name='x'), Var(name='y')), right=Const(value=12321)))), expr=Sum(Var(name='x'), Var(name='y')))))), b=Abs(arg=Sub(left=Const(value=111), right=Ref(name='_m')))), GCD(a=Abs(arg=Sub(left=Const(value=0), right=Ref(name='_n'))), b=Abs(arg=Sub(left=Const(value=0), right=Const(value=111))))),\n \"result\": Div(Sum(Ref(\"area_2x\"), Const(2), Mul(Const(-1), Ref(\"boundary\"))), Const(2)),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "ALG", "secondary_domain": "NT", "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "COUNT_SUM_EQUALS", "lemma_paths": ["COUNT_SUM_EQUALS", "COUNT_CARTESIAN", "B3"], "recipe_id": "a84f44", "seed_template_id": "geo_count_lattice_triangle_v1", "ending_id": null, "olympiad_level": 7, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["B3", "COUNT_CARTESIAN", "COUNT_SUM_EQUALS"], "num_lemmas": 3, "generation_time": 0.015, "created_at": "2026-02-08T08:59:08.965909Z", "verification": {"verified": true, "answer": 9546, "timestamp": "2026-02-08T08:59:08.980797Z"}, "problem_hash": "3ea163", "license": "CC BY 4.0", "llm_solvers": [{"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 9546, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 269, "completion_tokens": 1101}, "timestamp": "2026-02-13T23:02:20.598Z"}, {"id": 8, "model": "mathstral", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 346, "completion_tokens": 1002}, "timestamp": "2026-02-20T22:45:34.821Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "answer": 9415, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 350, "completion_tokens": 645}, "timestamp": "2026-02-12T02:22:58.809Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 9546, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 312, "completion_tokens": 1498}, "timestamp": "2026-02-16T18:23:37.911Z"}], "solution_status": 1, "lemma_applicability": {"B3": "ok", "COUNT_CARTESIAN": "ok", "COUNT_SUM_EQUALS": "ok", "C2": "no", "COUNT_COPRIME_GRID": "no", "COUNT_INTEGER_RANGE": "no", "V8": "no", "V8_SUM": "no"}, "irt_difficulty": {"lo": -5.55, "mid": -3.01, "hi": 0.32}}
{"id": "05c3dd_n", "alias": "comb_binomial_compute_v1_1218484723_6332", "problem": "A factory must cut a rectangular metal sheet of area $20736$ square centimeters into smaller identical rectangular tiles, where the side lengths (in centimeters) are positive integers. Among all such rectangles, consider those whose length plus width is as small as possible; let this minimum perimeter half-sum be $m = x + y$ for some optimal dimensions $(x, y)$ with $xy = 20736$.\n\nSeparately, an engineer examines all integer pairs $(a_1, b_1)$ with $1 \\le a_1 \\le 15$, $1 \\le b_1 \\le 15$, and $a_1 \\le b_1$ that satisfy\n$$2a_1^{2} - 4a_1b_1 + 2b_1^{2} = m,$$\nand calls the number of such pairs $C$.\n\nNow she designs a component using two parameters $(a, b)$, where $1 \\le a \\le 26$ and $1 \\le b \\le 26$. The cost of a design $(a, b)$ is\n$$b^{3} + 3ab^{2} + C a^{2}b.$$\nLet $k$ be the smallest possible cost over all allowed $(a, b)$, and define $Q = \\binom{16}{k}$. Compute $Q$. ", "answer": 11440, "graph": "", "domain": "COMB", "secondary_domain": null, "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "B3", "lemma_paths": ["B3/QF_PSD_ORBIT/POLY3_MIN"], "recipe_id": "ef01e1", "seed_template_id": "comb_binomial_compute_v1", "ending_id": null, "olympiad_level": 7, "variant": "narrative", "parent_id": "05c3dd", "lemma_set": ["B3", "POLY3_MIN", "QF_PSD_ORBIT"], "num_lemmas": 3, "generation_time": 0.013, "created_at": "2026-02-25T07:53:28.398271Z", "problem_hash": "2d3ab3", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "answer": 11440, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 398, "completion_tokens": 2957}, "timestamp": "2026-03-31T01:08:53.528Z"}, {"id": 2, "model": "openai/gpt-oss-120b", "answer": 11440, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 398, "completion_tokens": 2368}, "timestamp": "2026-05-03T14:37:41.778Z"}, {"id": 16, "model": "Qwen/Qwen3-Next-80B-A3B-Thinking", "answer": 11440, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 358, "completion_tokens": 9470}, "timestamp": "2026-03-31T21:08:20.745Z"}, {"id": 36, "model": "qwen2.5:3b-32k", "answer": 1, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 356, "completion_tokens": 1729}, "timestamp": "2026-04-23T06:07:54.658Z"}], "solution_status": 1, "lemma_applicability": {"B3": "ok", "POLY3_MIN": "ok_later", "QF_PSD_ORBIT": "ok_later", "COUNT_CARTESIAN": "no", "COUNT_COPRIME_GRID": "no", "V7": "no", "V8": "no", "V8_SUM": "no"}, "irt_difficulty": {"lo": -5.37, "mid": 0.23, "hi": 5.22}}
{"id": "8b9513", "alias": "nt_min_coprime_above_v1_717093673_691", "problem": "Let $T$ be the set of all positive integers $t$ such that $9 \\leq t \\leq 971$ and there exist positive integers $a$, $b$ with $1 \\leq a \\leq 14$, $1 \\leq b \\leq 183$, satisfying $t = 4a + 5b$.\n\nLet $m$ be the number of positive integers $n$ such that $1 \\leq n \\leq |T|$ and the sum of the decimal digits of $n$ is divisible by 2.\n\nLet $A$ be the set of integers $n_1$ such that $37249 < n_1 \\leq 37734$ and $\\gcd(n_1, m) = 1$. Let $r$ be the smallest element of $A$.\n\nCompute the remainder when $44121 \\cdot r$ is divided by 77975.", "answer": 72296, "graph": "graphs = [\n Graph(\n let={\n \"_m\": Const(77975),\n \"_n\": Const(2),\n \"start\": Const(37249),\n \"upper\": Const(37734),\n \"modulus\": CountOverSet(set=SolutionsSet(var=Var(\"n\"), condition=And(Geq(Var(\"n\"), Const(1)), Leq(Var(\"n\"), CountOverSet(set=SolutionsSet(var=Var(\"t\"), condition=Exists(var=Var(name='a'), condition=Exists(var=Var(name='b'), condition=And(Geq(left=Var(name='a'), right=Const(value=1)), Leq(left=Var(name='a'), right=Const(value=14)), Geq(left=Var(name='b'), right=Const(value=1)), Leq(left=Var(name='b'), right=Const(value=183)), Geq(left=Var(name='t'), right=Const(value=9)), Leq(left=Var(name='t'), right=Const(value=971)), Eq(left=Var(name='t'), right=Sum(Mul(Const(value=4), Var(name='a')), Mul(Const(value=5), Var(name='b')))))))))), Eq(Mod(value=DigitSum(Var(\"n\")), modulus=Ref(\"_n\")), Const(0))))),\n \"result\": MinOverSet(set=SolutionsSet(var=Var(\"n1\"), condition=And(Gt(Var(\"n1\"), Ref(\"start\")), Leq(Var(\"n1\"), Ref(\"upper\")), Eq(GCD(a=Var(\"n1\"), b=Ref(\"modulus\")), Const(1))))),\n \"Q\": Mod(value=Mul(Const(44121), Ref(\"result\")), modulus=Ref(\"_m\")),\n },\n goal=Ref(\"Q\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "EXTREMUM", "evaluator_id": "sympy", "root_lemma": "LIN_FORM", "lemma_paths": ["LIN_FORM/L3B"], "recipe_id": "db250f", "seed_template_id": "nt_min_coprime_above_v1", "ending_id": null, "olympiad_level": 7, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["L3B", "LIN_FORM"], "num_lemmas": 2, "generation_time": 0.118, "created_at": "2026-02-08T15:36:11.269503Z", "verification": {"verified": true, "answer": 72296, "timestamp": "2026-02-08T15:36:11.387066Z"}, "problem_hash": "d54e33", "license": "CC BY 4.0", "llm_solvers": [{"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 72296, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 232, "completion_tokens": 7269}, "timestamp": "2026-02-16T10:40:16.604Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 308, "completion_tokens": 474}, "timestamp": "2026-02-12T08:52:06.727Z"}], "solution_status": 1, "lemma_applicability": {"LIN_FORM": "ok", "L3B": "ok_later", "DS2": "no", "MAX_PRIME_BELOW": "no", "POLY_PADIC_VAL_CONST": "no", "V3": "no", "V7": "no"}, "irt_difficulty": {"lo": -7.08, "mid": -0.29, "hi": 6.49}}
{"id": "58318b", "alias": "algebra_quadratic_discriminant_v1_1218484723_5715", "problem": "Let $R$ be the number of ordered pairs $(a_1, b_1)$ of positive integers with $1 \\le a_1, b_1 \\le 40$ such that\n$$-68a_1b_1^{3} + 17a_1^{4} + 17b_1^{4} - 68a_1^{3}b_1 + 102a_1^{2}b_1^{2} = 4352.$$\nDefine\n$$D = -2^{2} - \\left|\\left\\{(a_2, b_2) : 1 \\le a_2, b_2 \\le 10,\\ a_2 \\le b_2,\\ C \\cdot b_2^{2} - 4a_2b_2 + 2a_2^{2} = R\\right\\}\\right| \\cdot 1 \\cdot 7,$$\nwhere\n$$C = \\min\\{98b_3^{3} - 96a_3b_3^{2} + 24a_3^{2}b_3 : 1 \\le a_3, b_3 \\le 26\\}.$$\nCompute\n$$2 \\cdot [D > 0] + [D = 0],$$\nwhere $[P]$ denotes $1$ if the statement $P$ is true and $0$ otherwise.", "answer": 0, "graph": "graphs = [\n Graph(\n let={\n \"_c\": Const(2),\n \"_m\": Const(3),\n \"_n\": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"a1\"), Var(\"b1\")]), condition=And(Geq(Var(\"a1\"), Const(1)), Leq(Var(\"a1\"), Const(40)), Geq(Var(\"b1\"), Const(1)), Leq(Var(\"b1\"), Const(40)), Eq(Sum(Mul(Const(-68), Var(\"a1\"), Pow(Var(\"b1\"), Const(3))), Mul(Const(17), Pow(Var(\"a1\"), Const(4))), Mul(Const(17), Pow(Var(\"b1\"), Const(4))), Mul(Const(-68), Pow(Var(\"a1\"), Const(3)), Var(\"b1\")), Mul(Const(102), Pow(Var(\"a1\"), Const(2)), Pow(Var(\"b1\"), Const(2)))), Const(4352))))),\n \"a\": Const(1),\n \"b\": Const(-2),\n \"c\": Const(7),\n \"D\": Sub(Pow(Ref(\"b\"), Const(2)), Mul(CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"a2\"), Var(\"b2\")]), condition=And(Geq(Var(\"a2\"), Const(1)), Leq(Var(\"a2\"), Const(10)), Geq(Var(\"b2\"), Const(1)), Leq(Var(\"b2\"), Const(10)), Leq(Var(\"a2\"), Var(\"b2\")), Eq(Sum(Mul(MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"a3\"), Var(\"b3\")]), condition=And(Geq(Var(\"a3\"), Const(1)), Leq(Var(\"a3\"), Const(26)), Geq(Var(\"b3\"), Const(1)), Leq(Var(\"b3\"), Const(26)))), expr=Sum(Mul(Const(98), Pow(Var(\"b3\"), Ref(\"_m\"))), Mul(Const(-96), Var(\"a3\"), Pow(Var(\"b3\"), Const(2))), Mul(Const(24), Pow(Var(\"a3\"), Ref(\"_c\")), Var(\"b3\"))))), Pow(Var(\"b2\"), Const(2))), Mul(Const(-4), Var(\"a2\"), Var(\"b2\")), Mul(Const(2), Pow(Var(\"a2\"), Const(2)))), Ref(\"_n\"))))), Ref(\"a\"), Ref(\"c\"))),\n \"result\": Sum(Mul(Const(2), Iverson(condition=Gt(Ref(\"D\"), Const(0)))), Iverson(condition=Eq(Ref(\"D\"), Const(0)))),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "ALG", "secondary_domain": null, "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "POLY4_COUNT", "lemma_paths": ["POLY4_COUNT/QF_PSD_ORBIT", "POLY3_MIN/QF_PSD_ORBIT"], "recipe_id": "42c797", "seed_template_id": "algebra_quadratic_discriminant_v1", "ending_id": null, "olympiad_level": 7, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["POLY3_MIN", "POLY4_COUNT", "QF_PSD_ORBIT"], "num_lemmas": 3, "generation_time": 0.31, "created_at": "2026-02-25T07:15:51.547399Z", "verification": {"verified": true, "answer": 0, "timestamp": "2026-02-25T07:15:51.857857Z"}, "problem_hash": "1d2fa4", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "answer": 0, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 421, "completion_tokens": 3357}, "timestamp": "2026-03-29T22:26:01.749Z"}, {"id": 2, "model": "openai/gpt-oss-120b", "answer": 0, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 421, "completion_tokens": 3600}, "timestamp": "2026-04-19T15:14:26.964Z"}, {"id": 16, "model": "Qwen/Qwen3-Next-80B-A3B-Thinking", "answer": 0, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 377, "completion_tokens": 11376}, "timestamp": "2026-03-30T13:07:25.172Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 2, "score": 1, "correct": {"strict": false, "boxed": false, "relaxed": true}, "usage": {"prompt_tokens": 382, "completion_tokens": 960}, "timestamp": "2026-04-19T23:51:18.239Z"}, {"id": 36, "model": "qwen2.5:3b-32k", "answer": 1, "score": 1, "correct": {"strict": false, "boxed": false, "relaxed": true}, "usage": {"prompt_tokens": 375, "completion_tokens": 1298}, "timestamp": "2026-04-21T20:28:58.966Z"}], "solution_status": 1, "lemma_applicability": {"POLY3_MIN": "ok", "POLY4_COUNT": "ok", "QF_PSD_ORBIT": "ok_later", "POLY_PADIC_VAL_CONST": "no"}, "irt_difficulty": {"lo": -3.33, "mid": 1.03, "hi": 5.26}}
{"id": "3c4c30", "alias": "alg_telescope_v1_1218484723_940", "problem": "Let\n$$T = \\sum_{k=0}^{931} \\left|\\left\\{ a : 0 \\le a \\le 72,\\ 3\\bigl(3(3a^{3} + a - 3 \\bmod 73)^{3} + (3a^{3} + a - 3 \\bmod M) - 3 \\bmod 73\\bigr)^{3} + \\bigl(3(3a^{3} + a - 3 \\bmod 73)^{3} + (3a^{3} + a - 3 \\bmod 73) - 3 \\bmod 73\\bigr) - 3 \\bmod 73 = a,\\\\ 3a^{3} + a - 3 \\bmod 73 \\ne a,\\\\ 3(3a^{3} + a - 3 \\bmod 73)^{3} + (3a^{3} + a - 3 \\bmod 73) - 3 \\bmod 73 \\ne a \\right\\}\\right| \\cdot k^{E} + 3k + 1 \\bmod 3304,$$\nwhere\n$$M = \\left|\\left\\{v : 128 \\le v \\le 15488,\\ \\text{there exist integers } a,b \\text{ with } 1 \\le a \\le 11,\\ 1 \\le b \\le 11 \\text{ such that } 18b^{2} + 50a^{2} + 60ab = v \\right\\}\\right|,$$\nand\n$$E = \\min\\{98b^{3} + 24a_1^{2}b - 96a_1b^{2} : (a_1,b),\\ 1 \\le a_1 \\le 17,\\ 1 \\le b \\le 17\\}.$$\nFind the remainder when $94709T$ is divided by $51493$.", "answer": 34670, "graph": "graphs = [\n Graph(\n let={\n \"_d\": Const(3),\n \"_m\": Const(51493),\n \"_n\": Const(3),\n \"result\": Mod(value=Summation(var=\"k\", start=Const(0), end=Const(931), expr=Sum(Mul(CountOverSet(set=SolutionsSet(var=Var(\"a\"), condition=And(Geq(Var(\"a\"), Const(0)), Leq(Var(\"a\"), Const(72)), Eq(Mod(value=Sum(Mul(Const(3), Pow(Mod(value=Sum(Mul(Const(3), Pow(Mod(value=Sum(Mul(Const(3), Pow(Var(\"a\"), Const(3))), Var(\"a\"), Const(-3)), modulus=Const(73)), Const(3))), Mod(value=Sum(Mul(Const(3), Pow(Var(\"a\"), Const(3))), Var(\"a\"), Const(-3)), modulus=CountOverSet(set=SolutionsSet(var=Var(\"v\"), condition=And(Geq(Var(\"v\"), Const(128)), Leq(Var(\"v\"), Const(15488)), Exists(var=Tuple(elements=[Var(name='a'), Var(name='b')]), condition=And(Geq(left=Var(name='a'), right=Const(value=1)), Leq(left=Var(name='a'), right=Const(value=11)), Geq(left=Var(name='b'), right=Const(value=1)), Leq(left=Var(name='b'), right=Const(value=11)), Eq(left=Sum(Mul(Const(value=18), Pow(base=Var(name='b'), exp=Const(value=2))), Mul(Const(value=50), Pow(base=Var(name='a'), exp=Const(value=2))), Mul(Const(value=60), Var(name='a'), Var(name='b'))), right=Var(name='v')))))))), Const(-3)), modulus=Const(73)), Const(3))), Mod(value=Sum(Mul(Const(3), Pow(Mod(value=Sum(Mul(Const(3), Pow(Var(\"a\"), Const(3))), Var(\"a\"), Const(-3)), modulus=Const(73)), Const(3))), Mod(value=Sum(Mul(Ref(\"_n\"), Pow(Var(\"a\"), Ref(\"_d\"))), Var(\"a\"), Const(-3)), modulus=Const(73)), Const(-3)), modulus=Const(73)), Const(-3)), modulus=Const(73)), Var(\"a\")), Neq(Mod(value=Sum(Mul(Const(3), Pow(Var(\"a\"), Const(3))), Var(\"a\"), Const(-3)), modulus=Const(73)), Var(\"a\")), Neq(Mod(value=Sum(Mul(Const(3), Pow(Mod(value=Sum(Mul(Const(3), Pow(Var(\"a\"), Const(3))), Var(\"a\"), Const(-3)), modulus=Const(73)), Const(3))), Mod(value=Sum(Mul(Const(3), Pow(Var(\"a\"), Const(3))), Var(\"a\"), Const(-3)), modulus=Const(73)), Const(-3)), modulus=Const(73)), Var(\"a\"))))), Pow(Var(\"k\"), MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var(\"a1\"), Var(\"b\")]), condition=And(Geq(Var(\"a1\"), Const(1)), Leq(Var(\"a1\"), Const(17)), Geq(Var(\"b\"), Const(1)), Leq(Var(\"b\"), Const(17)))), expr=Sum(Mul(Const(98), Pow(Var(\"b\"), Const(3))), Mul(Const(24), Pow(Var(\"a1\"), Const(2)), Var(\"b\")), Mul(Const(-96), Var(\"a1\"), Pow(Var(\"b\"), Const(2)))))))), Mul(Const(3), Var(\"k\")), Const(1))), modulus=Const(3304)),\n \"_c\": Const(94709),\n \"Q\": Mod(value=Mul(Ref(\"_c\"), Ref(\"result\")), modulus=Ref(\"_m\")),\n },\n goal=Ref(\"Q\"),\n )\n]", "domain": "ALG", "secondary_domain": null, "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "QF_PSD_DISTINCT", "lemma_paths": ["QF_PSD_DISTINCT/POLY_ORBIT_COUNT", "POLY3_MIN"], "recipe_id": "e681dd", "seed_template_id": "alg_telescope_v1", "ending_id": null, "olympiad_level": 8, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["POLY3_MIN", "POLY_ORBIT_COUNT", "QF_PSD_DISTINCT"], "num_lemmas": 3, "generation_time": 0.596, "created_at": "2026-02-25T02:38:02.435600Z", "verification": {"verified": true, "answer": 34670, "timestamp": "2026-02-25T02:38:03.031508Z"}, "problem_hash": "4c6654", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 540, "completion_tokens": 32768}, "timestamp": "2026-03-10T03:09:39.422Z"}, {"id": 2, "model": "openai/gpt-oss-120b", "answer": 22378, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 540, "completion_tokens": 8167}, "timestamp": "2026-04-18T23:27:52.755Z"}, {"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 0, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 463, "completion_tokens": 9318}, "timestamp": "2026-04-19T09:09:22.197Z"}, {"id": 10, "model": "qwen2-math:7b", "answer": 2688, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 519, "completion_tokens": 1400}, "timestamp": "2026-03-29T02:00:33.001Z"}, {"id": 16, "model": "Qwen/Qwen3-Next-80B-A3B-Thinking", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 524, "completion_tokens": 32768}, "timestamp": "2026-03-29T04:27:44.723Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 947, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 507, "completion_tokens": 636}, "timestamp": "2026-04-19T00:54:43.250Z"}, {"id": 36, "model": "qwen2.5:3b-32k", "answer": 0, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 522, "completion_tokens": 980}, "timestamp": "2026-04-21T09:08:28.365Z"}, {"id": 38, "model": "google/gemma-3-27b-it", "answer": 0, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 494, "completion_tokens": 1804}, "timestamp": "2026-04-20T01:21:53.101Z"}], "solution_status": 0, "lemma_applicability": {"POLY3_MIN": "ok", "QF_PSD_DISTINCT": "ok", "POLY_ORBIT_COUNT": "ok_later", "POLY_PADIC_VAL_CONST": "no"}, "irt_difficulty": {"lo": 5.81, "mid": 8.21, "hi": 10.0}}
{"id": "266639", "alias": "nt_count_with_divisor_count_v1_677425708_628", "problem": "Let $m=12$. Let $T$ be the set of all integers $n$ such that $1\\le n\\le 62220$ and $m$ divides the $n$th Fibonacci number $F_n$. Let $N$ be the number of elements in $T$.\n\nLet $D$ be the set of all nonnegative integers $j$ such that $0\\le j\\le 5185$ and\n\\[\\binom{N}{j} \\equiv 1 \\pmod{2}.\\]\nLet $d$ be the number of elements in $D$.\n\nLet $U$ be the set of all integers $n$ such that $1\\le n\\le 16384$ and the number of positive divisors of $n$ is equal to $d$. Let $R$ be the number of elements in $U$.\n\nCompute $R$.", "answer": 1421, "graph": "graphs = [\n Graph(\n let={\n \"_m\": Const(12),\n \"_n\": CountOverSet(set=SolutionsSet(var=Var(\"n\"), condition=And(Geq(Var(\"n\"), Const(1)), Leq(Var(\"n\"), Const(62220)), Divides(divisor=Ref(\"_m\"), dividend=Fibonacci(arg=Var(name='n')))))),\n \"upper\": Const(16384),\n \"div_count\": CountOverSet(set=SolutionsSet(var=Var(\"j\"), condition=And(Geq(Var(\"j\"), Const(0)), Leq(Var(\"j\"), Const(5185)), Eq(Mod(value=Binom(n=Ref(\"_n\"), k=Var(\"j\")), modulus=Const(2)), Const(1))), domain='nonnegative_integers')),\n \"result\": CountOverSet(set=SolutionsSet(var=Var(\"n\"), condition=And(Geq(Var(\"n\"), Const(1)), Leq(Var(\"n\"), Ref(\"upper\")), Eq(NumDivisors(n=Var(\"n\")), Ref(\"div_count\"))))),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "COUNT_FIB_DIVISIBLE", "lemma_paths": ["COUNT_FIB_DIVISIBLE/V8"], "recipe_id": "82a267", "seed_template_id": "nt_count_with_divisor_count_v1", "ending_id": null, "olympiad_level": 9, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["COUNT_FIB_DIVISIBLE", "V8"], "num_lemmas": 2, "generation_time": 0.672, "created_at": "2026-02-08T03:37:53.810180Z", "verification": {"verified": true, "answer": 1421, "timestamp": "2026-02-08T03:37:54.481970Z"}, "problem_hash": "54717d", "license": "CC BY 4.0", "llm_solvers": [{"id": 2, "model": "openai/gpt-oss-120b", "answer": 1503, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 299, "completion_tokens": 22060}, "timestamp": "2026-02-23T20:55:19.907Z"}, {"id": 4, "model": "NousResearch/Hermes-4-405B", "answer": 3, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 245, "completion_tokens": 577}, "timestamp": "2026-02-12T01:48:15.524Z"}, {"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 39, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 228, "completion_tokens": 4728}, "timestamp": "2026-02-11T14:54:10.842Z"}, {"id": 8, "model": "mathstral", "answer": 5186, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 262, "completion_tokens": 1369}, "timestamp": "2026-02-12T03:53:35.572Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 297, "completion_tokens": 558}, "timestamp": "2026-02-11T21:02:58.971Z"}, {"id": 15, "model": "Qwen/Qwen3-Coder-480B-A35B-Instruct", "answer": 140, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 253, "completion_tokens": 2110}, "timestamp": "2026-02-12T03:26:26.630Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct-fast", "answer": 512, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 265, "completion_tokens": 1396}, "timestamp": "2026-02-12T00:57:08.054Z"}], "solution_status": 0, "lemma_applicability": {"COUNT_FIB_DIVISIBLE": "ok", "V8": "ok_later", "COUNT_COPRIME_GRID": "no", "K15": "no", "MOD_MUL": "no", "V8_SUM": "no", "VAL_SUM_EQ": "no"}, "irt_difficulty": {"lo": 3.78, "mid": 6.08, "hi": 9.16}}
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