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c227af1 | 1 2 3 4 5 6 7 8 9 10 11 | {"id": "a3bdf6", "alias": "nt_sum_phi_v1_798873815_128", "problem": "Let $a = 163$ and $b = 132$. Let $d$ be a positive divisor of $\\gcd(a, b)$. Define $s = \\sum_{d \\mid \\gcd(a,b)} \\mu(d)$, where $\\mu$ is the Möbius function. Let $n = 187 \\cdot s$. Define $h = \\mu(n)^2$. Let $k$ be the number of positive integers less than or equal to 184815 that are divisible by 333. Let $u = k \\cdot h$. Compute the sum of $\\phi(m)$ for all positive integers $m$ from 1 to $u$, inclusive, where $\\phi$ is Euler's totient function.", "answer": 93734, "graph": "graphs = [\n Graph(\n let={\n \"_n\": Const(2),\n \"a\": Const(163),\n \"b\": Const(132),\n \"s\": SumOverDivisors(n=GCD(a=Ref(name='a'), b=Ref(name='b')), var='d', expr=MoebiusMu(n=Var(name='d'))),\n \"n\": Mul(Const(187), Ref(\"s\")),\n \"h\": Pow(MoebiusMu(n=Ref(name='n')), Ref(\"_n\")),\n \"upper\": Mul(CountOverSet(set=SolutionsSet(var=Var(\"k\"), condition=And(Geq(Var(\"k\"), Const(1)), Leq(Var(\"k\"), Const(184815)), Divides(divisor=Const(333), dividend=Var(\"k\"))), domain='positive_integers')), Ref(\"h\")),\n \"result\": SumOverSet(set=MapOverSet(set=SolutionsSet(var=Var(\"n\"), condition=And(Geq(Var(\"n\"), Const(1)), Leq(Var(\"n\"), Ref(\"upper\")))), expr=EulerPhi(n=Var(\"n\")))),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "SUM", "evaluator_id": "sympy", "root_lemma": "C2", "lemma_paths": ["C2/MOBIUS_SQUAREFREE", "MOBIUS_COPRIME"], "recipe_id": "87c621", "seed_template_id": "nt_sum_phi_v1", "ending_id": null, "olympiad_level": 6, "variant": "", "parent_id": "", "num_spawns": 2, "lemma_set": ["C2", "MOBIUS_COPRIME", "MOBIUS_SQUAREFREE"], "num_lemmas": 3, "generation_time": 0.03, "created_at": "2026-02-08T02:27:29.232688Z", "verification": {"verified": true, "answer": 93734, "timestamp": "2026-02-08T02:27:29.262504Z"}, "problem_hash": "e08c1a", "license": "CC BY 4.0", "llm_solvers": [{"id": 4, "model": "NousResearch/Hermes-4-405B", "answer": -59, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 207, "completion_tokens": 620}, "timestamp": "2026-02-08T20:40:08.584Z"}, {"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 95025, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 195, "completion_tokens": 4447}, "timestamp": "2026-02-09T00:07:18.155Z"}, {"id": 8, "model": "mathstral", "answer": 0, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 267, "completion_tokens": 365}, "timestamp": "2026-02-17T12:35:40.914Z"}], "solution_status": 0, "lemma_applicability": {"C2": "ok", "MOBIUS_COPRIME": "ok", "MOBIUS_SQUAREFREE": "ok_later", "COUNT_COPRIME_GRID": "no", "DS2": "no", "K18": "no", "L3b": "no", "LTE_SUM": "no"}, "irt_difficulty": {"lo": 3.02, "mid": 6.72, "hi": 10.0}}
{"id": "8a4734", "alias": "diophantine_fbi2_count_v1_1915831931_858", "problem": "Let $k = 720$. Determine the number of integers $d$ such that $2 \\leq d \\leq 122$, $d$ divides $k$, and $5 \\leq \\frac{k}{d} \\leq 125$.", "answer": 20, "graph": "graphs = [\n Graph(\n let={\n \"k\": Const(720),\n \"a\": Const(1),\n \"b\": Const(4),\n \"upper\": Const(121),\n \"result\": CountOverSet(set=SolutionsSet(var=Var(\"d\"), condition=And(Geq(Var(\"d\"), Const(2)), Leq(Var(\"d\"), Const(122)), Divides(divisor=Var(\"d\"), dividend=Ref(\"k\")), Geq(Div(Ref(\"k\"), Var(\"d\")), Const(5)), Leq(Div(Ref(\"k\"), Var(\"d\")), Const(125))))),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "C2", "lemma_paths": ["OMEGA_ONE", "B1/B3"], "recipe_id": "2bb35c", "seed_template_id": "diophantine_fbi2_count_v1", "ending_id": null, "olympiad_level": 4, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["B1", "B3", "C2", "OMEGA_ONE"], "num_lemmas": 4, "generation_time": 0.62, "created_at": "2026-02-08T15:43:21.435071Z", "verification": {"verified": true, "answer": 20, "timestamp": "2026-02-08T15:43:22.054734Z"}, "problem_hash": "0586f6", "license": "CC BY 4.0", "llm_solvers": [{"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 20, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 97, "completion_tokens": 1075}, "timestamp": "2026-02-16T12:16:44.363Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 160, "completion_tokens": 503}, "timestamp": "2026-02-12T09:03:17.599Z"}], "solution_status": 1, "lemma_applicability": {"B1": "ok", "OMEGA_ONE": "ok", "B3": "ok_later", "C2": "no", "K13": "no", "K16": "no", "V5": "no", "V7": "no"}, "irt_difficulty": {"lo": -6.89, "mid": 0.01, "hi": 6.91}}
{"id": "673f97", "alias": "comb_count_derangements_v1_677425708_4034", "problem": "Let $S$ be the set of all positive integers $p$ for which there exists an integer $q$ such that $pq = 108$, $\\gcd(p, q) = 1$, and $p < q$. Let $k$ be the number of elements in $S$. Let $m = 13013$, and let $n$ be the smallest divisor of $m$ that is at least $k$. Compute the number of derangements of $n$ elements, denoted $!n$.", "answer": 1854, "graph": "graphs = [\n Graph(\n let={\n \"_m\": Const(13013),\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=108)), 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 \"n\": MinOverSet(set=SolutionsSet(var=Var(\"d\"), condition=And(Geq(Var(\"d\"), Ref(\"_n\")), Divides(divisor=Var(\"d\"), dividend=Ref(\"_m\"))))),\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": "COPRIME_PAIRS", "lemma_paths": ["COPRIME_PAIRS/MIN_PRIME_FACTOR"], "recipe_id": "52cee2", "seed_template_id": "comb_count_derangements_v1", "ending_id": null, "olympiad_level": 6, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["COPRIME_PAIRS", "MIN_PRIME_FACTOR"], "num_lemmas": 2, "generation_time": 0.001, "created_at": "2026-02-08T06:24:14.207027Z", "verification": {"verified": true, "answer": 1854, "timestamp": "2026-02-08T06:24:14.208274Z"}, "problem_hash": "eb83fd", "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": 152, "completion_tokens": 1854}, "timestamp": "2026-02-12T23:45:46.975Z"}, {"id": 8, "model": "mathstral", "answer": 1854, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 208, "completion_tokens": 1019}, "timestamp": "2026-02-19T06:19:23.468Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "answer": 265, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 215, "completion_tokens": 754}, "timestamp": "2026-02-12T00:01:31.877Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 1854, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 184, "completion_tokens": 1308}, "timestamp": "2026-02-15T12:28:00.931Z"}], "solution_status": 1, "lemma_applicability": {"COPRIME_PAIRS": "ok", "MIN_PRIME_FACTOR": "ok_later", "COUNT_COPRIME_GRID": "no", "L3b": "no", "LTE_DIFF": "no", "V8": "no", "VAL_SUM_EQ": "no"}, "irt_difficulty": {"lo": -8.12, "mid": -4.91, "hi": -2.15}}
{"id": "3201e5", "alias": "geo_count_lattice_rect_v1_1918700295_4513", "problem": "Let $a = 361$ and $b = 167$. Define a lattice point as a point $(x, y)$ in the coordinate plane where both $x$ and $y$ are integers. Consider the rectangle consisting of all points $(x, y)$ such that $0 \\leq x \\leq a$ and $0 \\leq y \\leq b$. Let $N$ be the number of lattice points in this rectangle. Compute the remainder when $57118 \\cdot N$ is divided by $92463$.", "answer": 38304, "graph": "graphs = [\n Graph(\n let={\n \"a\": Const(361),\n \"b\": Const(167),\n \"result\": LatticePointsRect(a=Ref(name='a'), b=Ref(name='b')),\n \"Q\": Mod(value=Mul(Const(57118), Ref(\"result\")), modulus=Const(92463)),\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": 2, "variant": "", "parent_id": "", "num_spawns": 0, "generation_time": 0.002, "created_at": "2026-02-08T09:24:58.207839Z", "verification": {"verified": true, "answer": 38304, "timestamp": "2026-02-08T09:24:58.210200Z"}, "problem_hash": "9a28d7", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "answer": 38304, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 222, "completion_tokens": 3779}, "timestamp": "2026-02-24T11:18:10.608Z"}, {"id": 2, "model": "openai/gpt-oss-120b", "answer": 38304, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 222, "completion_tokens": 1870}, "timestamp": "2026-02-24T06:23:18.595Z"}, {"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 38304, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 154, "completion_tokens": 1570}, "timestamp": "2026-02-14T04:04:55.126Z"}, {"id": 8, "model": "mathstral", "answer": 1, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 210, "completion_tokens": 275}, "timestamp": "2026-02-21T05:25:54.324Z"}, {"id": 10, "model": "qwen2-math:7b", "answer": 6965, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 174, "completion_tokens": 700}, "timestamp": "2026-02-28T07:18:31.340Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 220, "completion_tokens": 201}, "timestamp": "2026-02-12T02:55:48.245Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 57138, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 188, "completion_tokens": 434}, "timestamp": "2026-02-24T03:43:43.707Z"}], "solution_status": 1, "irt_difficulty": {"lo": -2.25, "mid": 1.53, "hi": 5.27}}
{"id": "6be3d7", "alias": "nt_count_coprime_and_v1_677425708_1068", "problem": "Let $d$ range over the positive divisors of $\\gcd(12, 25)$. Define $L = \\sum_{d \\mid \\gcd(12,25)} \\mu(d)$, where $\\mu$ is the M\\\"obius function. Let $S$ be the set of all integers $n$ such that $L \\leq n \\leq 29032$, $\\gcd(n, 5) = 1$, and $\\gcd(n, 9) = 1$. Determine the number of elements in $S$.", "answer": 15484, "graph": "graphs = [\n Graph(\n let={\n \"upper\": Const(29032),\n \"k1\": Const(5),\n \"k2\": Const(9),\n \"result\": CountOverSet(set=SolutionsSet(var=Var(\"n\"), condition=And(Geq(Var(\"n\"), SumOverDivisors(n=GCD(a=Const(value=12), b=Const(value=25)), var='d', expr=MoebiusMu(n=Var(name='d')))), Leq(Var(\"n\"), Ref(\"upper\")), Eq(GCD(a=Var(\"n\"), b=Ref(\"k1\")), Const(1)), Eq(GCD(a=Var(\"n\"), b=Ref(\"k2\")), Const(1))))),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "MOBIUS_COPRIME", "lemma_paths": ["MOBIUS_COPRIME"], "recipe_id": "ac54ac", "seed_template_id": "nt_count_coprime_and_v1", "ending_id": null, "olympiad_level": 5, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["MOBIUS_COPRIME"], "num_lemmas": 1, "generation_time": 3.552, "created_at": "2026-02-08T03:59:43.104025Z", "verification": {"verified": true, "answer": 15484, "timestamp": "2026-02-08T03:59:46.655745Z"}, "problem_hash": "087372", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "answer": 15484, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 228, "completion_tokens": 2109}, "timestamp": "2026-02-09T15:22:57.032Z"}, {"id": 8, "model": "mathstral", "answer": 21406, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 217, "completion_tokens": 381}, "timestamp": "2026-02-18T07:24:15.902Z"}], "solution_status": 1, "lemma_applicability": {"MOBIUS_COPRIME": "ok", "LTE_DIFF_P2": "no", "LTE_SUM": "no", "MOD_ADD": "no", "V8_SUM": "no", "VAL_SUM_EQ": "no"}, "irt_difficulty": {"lo": -6.3, "mid": 0.12, "hi": 6.54}}
{"id": "261190", "alias": "modular_count_residue_v1_458359167_1190", "problem": "Let $r$ be the largest prime number such that $2 \\leq r \\leq 22$. Compute the number of positive integers $n$ such that $1 \\leq n \\leq 60000$ and $n \\equiv r \\pmod{21}$.", "answer": 2857, "graph": "graphs = [\n Graph(\n let={\n \"upper\": Const(60000),\n \"m\": Const(21),\n \"r\": MaxOverSet(set=SolutionsSet(var=Var(\"n\"), condition=And(Geq(Var(\"n\"), Const(2)), Leq(Var(\"n\"), Const(22)), IsPrime(Var(\"n\"))))),\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 },\n goal=Ref(\"result\"),\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": 4, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["MAX_PRIME_BELOW"], "num_lemmas": 1, "generation_time": 7.914, "created_at": "2026-02-08T04:29:09.674011Z", "verification": {"verified": true, "answer": 2857, "timestamp": "2026-02-08T04:29:17.587606Z"}, "problem_hash": "3b31e1", "license": "CC BY 4.0", "llm_solvers": [{"id": 8, "model": "mathstral", "answer": 2862, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 116, "completion_tokens": 378}, "timestamp": "2026-02-11T20:54:51.962Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "answer": 2857, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 163, "completion_tokens": 411}, "timestamp": "2026-02-11T20:52:55.310Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 2857, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 137, "completion_tokens": 417}, "timestamp": "2026-02-14T00:31:48.131Z"}], "solution_status": 1, "lemma_applicability": {"MAX_PRIME_BELOW": "ok", "C2": "no", "LTE_DIFF": "no", "POLY_PADIC_VAL_CONST": "no", "V3": "no", "V8_SUM": "no"}, "irt_difficulty": {"lo": -6.81, "mid": -4.47, "hi": -1.93}}
{"id": "f05546", "alias": "geo_count_lattice_triangle_v1_1218484723_4579", "problem": "Let $R = \\left|100 \\cdot \\sum_{k=1}^{16} \\varphi(k) \\cdot \\left\\lfloor \\frac{16}{k} \\right\\rfloor + 20 \\cdot (-9)\\right|$, \n$S = \\gcd(100, 9) + \\gcd(|20 - 100|, |136 - 9|) + \\gcd(|0 - 20|, |0 - N|)$ where $N = \\left|\\{ n : 1 \\le n \\le 680,\\ n \\equiv \\left\\lfloor \\frac{n}{2} \\right\\rfloor \\pmod{5} \\}\\right|$, and $T = \\frac{R + 2 - S}{2}$. Find the remainder when $76261T$ is divided by $89600$.", "answer": 32388, "graph": "graphs = [\n Graph(\n let={\n \"_m\": Const(136),\n \"_n\": Const(100),\n \"area_2x\": Abs(arg=Sum(Mul(Ref(name='_n'), Summation(expr=Mul(EulerPhi(n=Var(name='k')), Floor(arg=Div(left=Const(value=16), right=Var(name='k')))), var='k', start=Const(value=1), end=Const(value=16))), Mul(Const(value=20), Sub(left=Const(value=0), right=Const(value=9))))),\n \"boundary\": Sum(GCD(a=Abs(arg=Const(value=100)), b=Abs(arg=Const(value=9))), GCD(a=Abs(arg=Sub(left=Const(value=20), right=Const(value=100))), b=Abs(arg=Sub(left=Ref(name='_m'), right=Const(value=9)))), GCD(a=Abs(arg=Sub(left=Const(value=0), right=Const(value=20))), b=Abs(arg=Sub(left=Const(value=0), right=CountOverSet(set=SolutionsSet(var=Var(name='n'), condition=And(Geq(left=Var(name='n'), right=Const(value=1)), Leq(left=Var(name='n'), right=Const(value=680)), Congruent(a=Var(name='n'), b=Floor(arg=Div(left=Var(name='n'), right=Const(value=2))), modulus=Const(value=5))))))))),\n \"result\": Div(Sum(Ref(\"area_2x\"), Const(2), Mul(Const(-1), Ref(\"boundary\"))), Const(2)),\n \"Q\": Mod(value=Mul(Const(76261), Ref(\"result\")), modulus=Const(89600)),\n },\n goal=Ref(\"Q\"),\n )\n]", "domain": "GEOM", "secondary_domain": "NT", "goal": "COUNT", "evaluator_id": "sympy", "root_lemma": "L3C", "lemma_paths": ["L3C", "K2"], "recipe_id": "c92cf4", "seed_template_id": "geo_count_lattice_triangle_v1", "ending_id": null, "olympiad_level": 6, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["K2", "L3C"], "num_lemmas": 2, "generation_time": 0.01, "created_at": "2026-02-25T06:15:28.923052Z", "verification": {"verified": true, "answer": 32388, "timestamp": "2026-02-25T06:15:28.932977Z"}, "problem_hash": "976cf1", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "answer": 32388, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 300, "completion_tokens": 32768}, "timestamp": "2026-03-29T16:24:37.268Z"}, {"id": 2, "model": "openai/gpt-oss-120b", "answer": 32388, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 300, "completion_tokens": 1868}, "timestamp": "2026-04-19T11:09:46.355Z"}, {"id": 16, "model": "Qwen/Qwen3-Next-80B-A3B-Thinking", "answer": 32388, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 262, "completion_tokens": 15564}, "timestamp": "2026-03-30T05:53:53.306Z"}], "solution_status": 2, "lemma_applicability": {"K2": "ok", "L3C": "ok"}, "irt_difficulty": {"lo": -10.0, "mid": -2.19, "hi": 5.61}}
{"id": "2e2052", "alias": "nt_count_primes_v1_1742523217_5366", "problem": "Let $c$ be the number of positive integers $p$ for which there exists a positive integer $q$ such that $p \\cdot q = 54$, $\\gcd(p, q) = 1$, and $p < q$. Compute the number of prime numbers $n$ such that $c \\leq n \\leq 40000$.", "answer": 4203, "graph": "graphs = [\n Graph(\n let={\n \"upper\": Const(40000),\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=54)), 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": 4, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["COPRIME_PAIRS"], "num_lemmas": 1, "generation_time": 3.855, "created_at": "2026-02-08T10:56:28.977851Z", "verification": {"verified": true, "answer": 4203, "timestamp": "2026-02-08T10:56:32.832577Z"}, "problem_hash": "813097", "license": "CC BY 4.0", "llm_solvers": [{"id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "answer": 4203, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 119, "completion_tokens": 1762}, "timestamp": "2026-02-14T09:41:00.290Z"}, {"id": 8, "model": "mathstral", "answer": 1227, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 171, "completion_tokens": 423}, "timestamp": "2026-02-21T11:57:43.317Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 180, "completion_tokens": 361}, "timestamp": "2026-02-12T03:33:15.805Z"}], "solution_status": 1, "lemma_applicability": {"COPRIME_PAIRS": "ok", "K18": "no", "L3b": "no", "LTE_SUM": "no", "MOD_MUL": "no", "V8_SUM": "no"}, "irt_difficulty": {"lo": -4.94, "mid": 0.63, "hi": 6.92}}
{"id": "99ac19", "alias": "lte_diff_endings_v1_677425708_763", "problem": "Let $a = 84$, $b = 9$, $p = 5$, and $n = 225$. Compute the largest integer $k$ such that $p^k$ divides $a^n - b^n$. Multiply this $k$ by $16216$, and compute the remainder when the result is divided by $99325$.", "answer": 64864, "graph": "graphs = [\n Graph(\n let={\n \"a_val\": Const(84),\n \"b_val\": Const(9),\n \"p_val\": Const(5),\n \"n_val\": Const(225),\n \"a_pow\": Pow(Ref(\"a_val\"), Ref(\"n_val\")),\n \"b_pow\": Pow(Ref(\"b_val\"), Ref(\"n_val\")),\n \"pow_diff\": Sub(Ref(\"a_pow\"), Ref(\"b_pow\")),\n \"_inner_result\": MaxKDivides(target=Ref(\"pow_diff\"), base=Ref(\"p_val\")),\n \"_scale_k\": Const(16216),\n \"_scaled\": Mul(Ref(\"_scale_k\"), Ref(\"_inner_result\")),\n \"_mod_M\": Const(99325),\n \"x\": Mod(value=Ref(\"_scaled\"), modulus=Ref(\"_mod_M\")),\n },\n goal=Ref(\"x\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "LTE_DIFF", "lemma_paths": ["LTE_DIFF"], "recipe_id": "cf8260", "seed_template_id": "lte_diff_endings_v1", "ending_id": null, "olympiad_level": 5, "variant": "", "parent_id": "", "lemma_set": ["LTE_DIFF"], "num_lemmas": 1, "generation_time": 0.001, "created_at": "2026-02-08T03:43:15.508274Z", "verification": {"verified": true, "answer": 64864, "timestamp": "2026-02-08T03:43:15.509097Z"}, "problem_hash": "735273", "license": "CC BY 4.0", "llm_solvers": [{"id": 1, "model": "openai/gpt-oss-20b", "answer": 64864, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 188, "completion_tokens": 763}, "timestamp": "2026-02-08T21:04:31.187Z"}, {"id": 8, "model": "mathstral", "answer": 615, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 178, "completion_tokens": 550}, "timestamp": "2026-02-18T06:20:21.249Z"}], "solution_status": 1, "lemma_applicability": {"LTE_DIFF": "ok", "COUNT_PRIMES": "no", "MOD_MUL": "no", "POLY_PADIC_VAL_CONST": "no", "V7": "no", "V8": "no"}, "irt_difficulty": {"lo": -6.3, "mid": 0.12, "hi": 6.54}}
{"id": "40743e", "alias": "algebra_poly_eval_v1_153355830_1633", "problem": "Let $x = 11$. Compute the value of $2x^2 - 7x + \\sum_{k=1}^{3} k$.", "answer": 171, "graph": "graphs = [\n Graph(\n let={\n \"x\": Const(11),\n \"result\": Sum(Mul(Const(2), Pow(Ref(\"x\"), Const(2))), Mul(Const(-7), Ref(\"x\")), Summation(var=\"k\", start=Const(1), end=Const(3), expr=Var(\"k\"))),\n },\n goal=Ref(\"result\"),\n )\n]", "domain": "NT", "secondary_domain": null, "goal": "COMPUTE", "evaluator_id": "sympy", "root_lemma": "B3", "lemma_paths": ["SUM_ARITHMETIC"], "recipe_id": "eb34f0", "seed_template_id": "algebra_poly_eval_v1", "ending_id": null, "olympiad_level": 2, "variant": "", "parent_id": "", "num_spawns": 0, "lemma_set": ["B3", "SUM_ARITHMETIC"], "num_lemmas": 2, "generation_time": 0.015, "created_at": "2026-02-08T06:32:07.733279Z", "verification": {"verified": true, "answer": 171, "timestamp": "2026-02-08T06:32:07.748248Z"}, "problem_hash": "ae4b61", "license": "CC BY 4.0", "llm_solvers": [{"id": 8, "model": "mathstral", "answer": 169, "score": 0, "correct": {"strict": false, "boxed": false, "relaxed": false}, "usage": {"prompt_tokens": 88, "completion_tokens": 239}, "timestamp": "2026-02-15T17:34:43.149Z"}, {"id": 11, "model": "google/gemma-2-9b-it", "answer": 171, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 136, "completion_tokens": 184}, "timestamp": "2026-02-12T00:13:11.067Z"}, {"id": 17, "model": "meta-llama/Llama-3.3-70B-Instruct", "answer": 171, "score": 3, "correct": {"strict": true, "boxed": true, "relaxed": true}, "usage": {"prompt_tokens": 110, "completion_tokens": 263}, "timestamp": "2026-02-15T15:01:18.111Z"}], "solution_status": 1, "lemma_applicability": {"SUM_ARITHMETIC": "ok", "COUNT_FIB_DIVISIBLE": "no", "L3b": "no", "MAX_PRIME_BELOW": "no", "MOD_FACTORIAL": "no", "VAL_SUM_EQ": "no"}, "irt_difficulty": {"lo": -8.94, "mid": -5.85, "hi": -3.53}}
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