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8859d7
algebra_poly_eval_v1_784195855_2589
Let $$ A = \sum_{k=1}^{3} \phi(k) \left\lfloor \frac{3}{k} \right\rfloor,$$ where $\phi$ denotes Euler's totient function. Define $$ x = \frac{A \cdot 6^5 - 67 \cdot 6^4 + 20 \cdot 6^3 - 95 \cdot 6^2 - 35 \cdot 6 - 99}{-5}.$$ Compute the value of $Q = |x|$.
7,917
graphs = [ Graph( let={ "_n": Const(3), "x": Const(6), "result": Div(Sum(Mul(Summation(var="k", start=Const(1), end=Const(3), expr=Mul(EulerPhi(n=Var("k")), Floor(Div(Ref("_n"), Var("k"))))), Pow(Ref("x"), Const(5))), Mul(Const(-67), Pow(Ref("x"), Const(4))), Mul(Const(20...
NT
null
COMPUTE
sympy
K2
[ "K2" ]
6897ab
algebra_poly_eval_v1
null
4
0
[ "K2" ]
1
0.003
2026-02-08T05:53:53.302390Z
{ "verified": true, "answer": 7917, "timestamp": "2026-02-08T05:53:53.304897Z" }
46a2fe
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 157, "completion_tokens": 810 }, "timestamp": "2026-02-12T16:06:03.156Z", "answer": 7917 }, { ...
1
[ { "lemma": "K15", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "K2", "status": "ok" }, { "lemma": "L3b", "status": "no" }, { "lemma": "MAX_VAL", "status": "no" }, { "lemma": "MOD_ADD", "status": "no" } ]
{ "lo": -3.51, "mid": 1.33, "hi": 6.56 }
5caf63
lin_form_endings_v1_655260480_6147
Let $a = 28$ and $b = 70$. Define $k = 413$ and let $d = \gcd(a, b)$. Compute $m = \left\lfloor \frac{k}{\gcd(k, d)} \right\rfloor$. Let $s = 14728 \cdot m$. Find the remainder when $s$ is divided by 58819.
45,486
graphs = [ Graph( let={ "a_coeff": Const(28), "b_coeff": Const(70), "k_val": Const(413), "step": GCD(a=Ref("a_coeff"), b=Ref("b_coeff")), "_inner_result": Floor(Div(Ref("k_val"), GCD(a=Ref("k_val"), b=Ref("step")))), "_scale_k": Const(1...
COMB
NT
COMPUTE
sympy
LIN_FORM
[ "LIN_FORM" ]
7b2633
lin_form_endings_v1
null
4
null
[ "LIN_FORM" ]
1
0.001
2026-02-08T18:51:27.935664Z
{ "verified": true, "answer": 45486, "timestamp": "2026-02-08T18:51:27.936515Z" }
a2d65f
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 126, "completion_tokens": 1260 }, "timestamp": "2026-02-18T20:03:27.563Z", "answer": 45486 }, ...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8_SUM", "status": "no" ...
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
bd05d5
modular_sum_quadratic_residues_v1_349078426_1736
Let $p$ be the number of ordered pairs of positive odd integers $(x_1, x_2)$ such that $x_1 + x_2 = 482$. Compute $$ \frac{p(p-1)}{4}. $$
14,460
graphs = [ Graph( let={ "_n": Const(4), "p": 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")), Con...
NT
null
SUM
sympy
COMB1
[ "COMB1" ]
567f58
modular_sum_quadratic_residues_v1
null
3
0
[ "COMB1" ]
1
0.002
2026-02-08T13:53:53.301413Z
{ "verified": true, "answer": 14460, "timestamp": "2026-02-08T13:53:53.303725Z" }
8530b4
CC BY 4.0
[ { "id": 8, "model": "mathstral", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 112, "completion_tokens": 541 }, "timestamp": "2026-02-16T05:09:34.844Z", "answer": 14460 }, { "id": 11, ...
2
[ { "lemma": "COMB1", "status": "ok" }, { "lemma": "K18", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" }, { "lemma": "V1", "status": "no" }, { "lemma": "V5", "status": "no" } ]
{ "lo": -10, "mid": -7.27, "hi": -4.54 }
d95d08
comb_count_partitions_v1_1470522791_258
Let $m = 6$ and $n = 9$. Define $S$ as the set of all ordered pairs $(x, y)$ of positive integers such that $x + y = m$. Let $P$ be the set of all products $xy$ where $(x, y) \in S$. Let $N = \sum_{k=1}^{\max P} \phi(k) \left\lfloor \frac{n}{k} \right\rfloor$, where $\phi$ denotes Euler's totient function. Compute the ...
89,134
graphs = [ Graph( let={ "_m": Const(6), "_n": Const(9), "n": Summation(var="k", start=Const(1), end=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(...
NT
COMB
COUNT
sympy
B1
[ "B1/K2" ]
ebd04c
comb_count_partitions_v1
null
5
0
[ "B1", "K2" ]
2
0.003
2026-02-08T12:55:51.112683Z
{ "verified": true, "answer": 89134, "timestamp": "2026-02-08T12:55:51.116147Z" }
610443
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 172, "completion_tokens": 1085 }, "timestamp": "2026-02-15T07:29:11.850Z", "answer": 89134 }, ...
1
[ { "lemma": "B1", "status": "ok" }, { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "K13", "status": "no" }, { "lemma": "K2", "status": "ok_later" }, { "lemma": "LTE_SUM", "status": "no" }, { ...
{ "lo": -5.14, "mid": 0.32, "hi": 6.51 }
2ce4b6
comb_bell_compute_v1_397696148_2830
Let $n$ be the number of nonnegative integers $j$ such that $0 \leq j \leq 36865$ and the binomial coefficient $\binom{36865}{j}$ is odd. Compute the Bell number $B_n$, which counts the number of partitions of a set of $n$ elements. Find the value of $15753 - B_n$.
11,613
graphs = [ Graph( let={ "_n": Const(36865), "n": CountOverSet(set=SolutionsSet(var=Var("j"), condition=And(Geq(Var("j"), Const(0)), Leq(Var("j"), Ref("_n")), Eq(Mod(value=Binom(n=Const(36865), k=Var("j")), modulus=Const(2)), Const(1))), domain='nonnegative_integers')), "r...
COMB
null
COMPUTE
sympy
V8
[ "V8" ]
86348e
comb_bell_compute_v1
null
7
0
[ "V8" ]
1
0.001
2026-02-08T14:06:12.921691Z
{ "verified": true, "answer": 11613, "timestamp": "2026-02-08T14:06:12.922808Z" }
81b410
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 190, "completion_tokens": 614 }, "timestamp": "2026-02-24T19:50:36.416Z", "answer": 11613 }, { "i...
1
[ { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "K14", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "V8", "status": "ok" }, { "lemma": "VAL_SUM_EQ", "status": "no" ...
{ "lo": -2.43, "mid": 1.2, "hi": 4.84 }
c0ce9a
nt_min_with_divisor_count_v1_865884756_4965
Let $n$ be the smallest positive integer such that $n \leq 33856$ and the number of positive divisors of $n$ is exactly 8. Find the value of $n$.
24
graphs = [ Graph( let={ "upper": Const(33856), "div_count": Const(8), "result": MinOverSet(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"))))), }, goal=Ref("res...
NT
null
EXTREMUM
sympy
C2
[ "VIETA_SUM", "ONE_PHI_1" ]
92db82
nt_min_with_divisor_count_v1
null
4
0
[ "C2", "ONE_PHI_1", "VIETA_SUM" ]
3
4.418
2026-02-08T18:18:06.879787Z
{ "verified": true, "answer": 24, "timestamp": "2026-02-08T18:18:11.298048Z" }
50f6a5
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 88, "completion_tokens": 1268 }, "timestamp": "2026-02-18T15:53:29.594Z", "answer": 24 }, { ...
1
[ { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "K18", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MOD_ADD", "status": "no" }, { "lemma": "ONE_PHI_1", "status": "ok" }, { ...
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
be6749
comb_factorial_compute_v1_1419126231_114
Let $n$ be the minimum value of $19b^3 - 15a^2b - 33ab^2 + 37a^3$ over all ordered pairs $(a, b)$ of positive integers with $1 \leq a \leq 23$ and $1 \leq b \leq \left|\{ v \geq 1 : v \leq 689 \text{ and } v = 41b'^2 - 74ab' + 34a'^2 \text{ for some integers } a', b' \in [1,5] \}\right|$. Let $Q = n!$. Compute $Q$.
40,320
graphs = [ Graph( let={ "_m": Const(2), "_n": Const(37), "n": MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var("a"), Var("b")]), condition=And(Geq(Var("a"), Const(1)), Leq(Var("a"), Const(23)), Geq(Var("b"), Const(1)), Leq(Var("b"), CountOverSet(set=Solu...
COMB
null
COMPUTE
sympy
QF_PSD_DISTINCT
[ "QF_PSD_DISTINCT/POLY3_MIN" ]
77024c
comb_factorial_compute_v1
null
5
0
[ "POLY3_MIN", "QF_PSD_DISTINCT" ]
2
0.004
2026-02-25T09:38:40.695116Z
{ "verified": true, "answer": 40320, "timestamp": "2026-02-25T09:38:40.698712Z" }
e72223
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 253, "completion_tokens": 7316 }, "timestamp": "2026-03-30T07:06:32.086Z", "answer": 40320 }, { "...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "POLY3_MIN", "status": "ok_later" }, { "lemma": "QF_PSD_DI...
{ "lo": -5.37, "mid": 0.23, "hi": 5.22 }
f8bf95
nt_count_gcd_equals_v1_458359167_2629
Let $n = 2$ and $k = 318$. Let $d$ be the smallest integer greater than or equal to $n$ that divides $8783743$. Let $U = 17711$. Let $r$ be the number of positive integers $n'$ such that $1 \leq n' \leq U$ and $\gcd(n', k) = d$. Compute the remainder when $44121 \cdot r$ is divided by $59153$.
46,885
graphs = [ Graph( let={ "_n": Const(2), "upper": Const(17711), "k": Const(318), "d": MinOverSet(set=SolutionsSet(var=Var("d"), condition=And(Geq(Var("d"), Ref("_n")), Divides(divisor=Var("d"), dividend=Const(8783743))))), "result": CountOverSet(set...
NT
null
COUNT
sympy
MIN_PRIME_FACTOR
[ "MIN_PRIME_FACTOR" ]
bc3776
nt_count_gcd_equals_v1
null
3
0
[ "MIN_PRIME_FACTOR" ]
1
1.357
2026-02-08T06:23:01.646073Z
{ "verified": true, "answer": 46885, "timestamp": "2026-02-08T06:23:03.002676Z" }
0b0df0
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 146, "completion_tokens": 1831 }, "timestamp": "2026-02-13T03:14:42.073Z", "answer": 46885 }, ...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "K13", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" }, { "lemma": "MIN_PRIME_FACTOR", "status": "ok" }, { "lemma": "MOD_MUL", "status": "no" }, { "lemma": "V7", "status": "no" } ]
{ "lo": -3.44, "mid": 1.34, "hi": 6.56 }
9f7031
v1_endings_v1_677425708_1121
Let $ n = 66487 $. Let $ v_5(n!) $ denote the largest integer $ k $ such that $ 5^k $ divides $ n! $, and let $ v_7(n!) $ denote the largest integer $ k $ such that $ 7^k $ divides $ n! $. Let $ a $ be the remainder when $ v_5(n!) $ is divided by 1000, and let $ b $ be the remainder when $ v_7(n!) $ is divided by 100. ...
61,877
graphs = [ Graph( let={ "n_val": Const(66487), "p1_val": Const(5), "p2_val": Const(7), "n_fact": Factorial(Ref("n_val")), "vp1": MaxKDivides(target=Ref("n_fact"), base=Ref("p1_val")), "vp2": MaxKDivides(target=Ref("n_fact"), base=Ref("p...
NT
null
COMPUTE
sympy
V1
[ "V1" ]
dae96f
v1_endings_v1
null
5
null
[ "V1" ]
1
0
2026-02-08T04:00:17.496400Z
{ "verified": true, "answer": 61877, "timestamp": "2026-02-08T04:00:17.496877Z" }
0d9b9f
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 227, "completion_tokens": 1245 }, "timestamp": "2026-02-09T15:56:41.903Z", "answer": 61877 }, { "...
1
[ { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "K14", "status": "no" }, { "lemma": "K15", "status": "no" }, { "lemma": "K5", "status": "same_pattern_wrong" }, { "lemma": "MOD_ADD", "statu...
{ "lo": -6.5, "mid": -0.2, "hi": 6.11 }
1b98a4
modular_mod_compute_v1_1125832087_1290
Compute the remainder when $-23104$ is divided by $14641$.
6,178
graphs = [ Graph( let={ "a": Const(-23104), "m": Const(14641), "result": Mod(value=Ref("a"), modulus=Ref("m")), }, goal=Ref("result"), ) ]
NT
null
COMPUTE
sympy
LIN_FORM
[ "LIN_FORM" ]
7b2633
modular_mod_compute_v1
null
2
0
[ "LIN_FORM" ]
1
0.009
2026-02-08T03:40:29.387861Z
{ "verified": true, "answer": 6178, "timestamp": "2026-02-08T03:40:29.396841Z" }
254528
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 126, "completion_tokens": 468 }, "timestamp": "2026-02-10T15:22:45.003Z", "answer": 6178 }, { "id...
1
[ { "lemma": "K14", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "L3c", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" }, { "lemma": "VAL_SUM_EQ", "status": "no" } ]
{ "lo": -7.01, "mid": -4.68, "hi": -2.15 }
ea31b6
diophantine_sum_product_min_v1_48377204_1601
Let $S = 26$. Let $P$ be the number of integers $t$ such that $10 \leq t \leq 186$ and there exist positive integers $a$ and $b$ with $1 \leq a \leq 48$, $1 \leq b \leq 6$, and $t = 3a + 7b$. Let $M$ be the maximum value of $xy$ over all pairs of positive integers $(x, y)$ such that $x + y = 10$. Determine the value of...
11
graphs = [ Graph( let={ "S": Const(26), "P": 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=48)), Geq(left=Var(n...
NT
null
EXTREMUM
sympy
LIN_FORM
[ "LIN_FORM", "B1" ]
2f9b70
diophantine_sum_product_min_v1
null
6
0
[ "B1", "LIN_FORM" ]
2
0.014
2026-02-08T16:15:13.091032Z
{ "verified": true, "answer": 11, "timestamp": "2026-02-08T16:15:13.105053Z" }
766a12
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 189, "completion_tokens": 5497 }, "timestamp": "2026-02-16T23:33:56.760Z", "answer": 11 }, { ...
1
[ { "lemma": "B1", "status": "ok" }, { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "K18", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "MOD_MUL", "status": "no" }, { "lemma": "V3", "status": "no" }, { ...
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
4f14a9
nt_gcd_compute_v1_168721529_1711
Let $a = 986193$ and $b = 1593081$. Define $d = \gcd(a, b)$. Let $\phi(n)$ denote Euler's totient function and $\tau(n)$ denote the number of positive divisors of $n$. Compute the remainder when $$ d + \phi\left(|d| + \phi(1)\right) + \tau\left(|d| + \phi(2)\right) $$ is divided by $84396$.
28,865
graphs = [ Graph( let={ "a": Const(986193), "b": Const(1593081), "result": GCD(a=Ref("a"), b=Ref("b")), "Q": Mod(value=Sum(Ref("result"), EulerPhi(n=Sum(Abs(arg=Ref(name='result')), EulerPhi(n=Const(1)))), NumDivisors(n=Sum(Abs(arg=Ref(name='result')), EulerPh...
NT
null
COMPUTE
sympy
ONE_PHI_1
[ "ONE_PHI_1", "ONE_PHI_2" ]
a76f7e
nt_gcd_compute_v1
null
3
0
[ "ONE_PHI_1", "ONE_PHI_2" ]
2
0.004
2026-02-08T13:53:29.536287Z
{ "verified": true, "answer": 28865, "timestamp": "2026-02-08T13:53:29.540416Z" }
56e0f2
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 215, "completion_tokens": 1809 }, "timestamp": "2026-02-09T20:39:42.282Z", "answer": 28865 }, { "...
1
[ { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "K14", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "ONE_PHI_1", "stat...
{ "lo": -6.5, "mid": -0.2, "hi": 6.11 }
80dbfd
nt_count_divisible_v1_1742523217_5512
Let $m$ be the number of positive integers $p$ for which there exists a positive integer $q$ such that \[pq=6,\quad \gcd(p,q)=1,\quad\text{and}\quad p<q.\] Let $n=337$, let $U=81796$, and let $d=11$. Let $R$ be the number of integers $t$ with $1\le t\le U$ such that $d\mid t$. Let $M$ be the smallest integer $k$ with...
44,177
graphs = [ Graph( let={ "_c": Const(2008), "_m": 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=6)), Eq(left=GCD(a=Var(name='p'), b=Var(name=...
NT
null
COUNT
sympy
COPRIME_PAIRS
[ "COPRIME_PAIRS/MIN_PRIME_FACTOR", "MAX_PRIME_BELOW" ]
aedac7
nt_count_divisible_v1
two_moduli
6
0
[ "COPRIME_PAIRS", "MAX_PRIME_BELOW", "MIN_PRIME_FACTOR" ]
3
3.248
2026-02-08T11:02:18.574697Z
{ "verified": true, "answer": 44177, "timestamp": "2026-02-08T11:02:21.822974Z" }
8441cb
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 215, "completion_tokens": 1281 }, "timestamp": "2026-02-14T10:15:56.196Z", "answer": 44177 }, ...
1
[ { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "K5", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok" }, { "lemma": "MAX_VAL", "status": "no" }, { "lemma": "MIN_PRIME_FACTOR", "status"...
{ "lo": -5.14, "mid": 0.32, "hi": 6.51 }
99d3e5
lin_form_endings_v1_1742523217_3510
Let $a = 40$ and $b = 70$. Define $d = \gcd(a, b)$. Let $k = 64$ and compute $g = \gcd(k, d)$. Define $r = \left\lfloor \frac{k}{g} \right\rfloor$. Multiply $r$ by $9227$, and let the result be $s$. Compute the remainder when $s$ is divided by $77410$.
63,034
graphs = [ Graph( let={ "a_coeff": Const(40), "b_coeff": Const(70), "k_val": Const(64), "step": GCD(a=Ref("a_coeff"), b=Ref("b_coeff")), "_inner_result": Floor(Div(Ref("k_val"), GCD(a=Ref("k_val"), b=Ref("step")))), "_scale_k": Const(92...
COMB
NT
COMPUTE
sympy
LIN_FORM
[ "LIN_FORM" ]
7b2633
lin_form_endings_v1
null
3
null
[ "LIN_FORM" ]
1
0.001
2026-02-08T05:55:29.672459Z
{ "verified": true, "answer": 63034, "timestamp": "2026-02-08T05:55:29.672977Z" }
13e116
CC BY 4.0
[ { "id": 8, "model": "mathstral", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 164, "completion_tokens": 393 }, "timestamp": "2026-02-11T23:21:53.941Z", "answer": 63862 }, { "id": 11...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", "status": "no" }, { "lemma": "V8_SUM", "status": "no" } ]
{ "lo": -7.01, "mid": -4.68, "hi": -2.15 }
2b71e4
nt_lcm_compute_v1_971394319_805
Let $a$ be the minimum value of $x + y$ over all ordered pairs of positive integers $(x, y)$ such that $xy = 2193361$. Let $b = 1971$. Define $L$ to be the least common multiple of $a$ and $b$. Let $c = 6000$ and $n = 2$. Compute the sum $\sum_{i=0}^{d-1} \left( \text{the } i\text{-th decimal digit of } L \right) \cdot...
6,747
graphs = [ Graph( let={ "_n": Const(2), "a": 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(2193361)))), expr=Sum(Var("x"), Var("y")))),...
NT
null
COMPUTE
sympy
B3
[ "B3" ]
0cd20d
nt_lcm_compute_v1
null
5
0
[ "B3" ]
1
0.002
2026-02-08T13:19:09.790976Z
{ "verified": true, "answer": 6747, "timestamp": "2026-02-08T13:19:09.793141Z" }
900dfa
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 202, "completion_tokens": 1863 }, "timestamp": "2026-02-15T12:54:45.716Z", "answer": 6747 }, {...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MOD_MUL", "status": "no" } ]
{ "lo": -5.14, "mid": 0.32, "hi": 6.51 }
88cd3f_n
comb_count_surjections_v1_1218484723_7021
A game show has 5 distinct prizes to distribute among 4 identical bonus chests, with no chest left empty. The number of ways to assign the prizes is $S(5,4)$, and the total configuration count is scaled by $4!$ to account for chest labeling. Let $M$ be this total. The final prize multiplier is the remainder when $41479...
70,800
COMB
null
COUNT
sympy
STARS_BARS
[ "STARS_BARS", "ONE_FACTORIAL_0" ]
71b4f8
comb_count_surjections_v1
null
2
null
[ "ONE_FACTORIAL_0", "STARS_BARS" ]
2
0.008
2026-02-25T08:26:40.229395Z
null
8877cf
88cd3f
narrative
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 204, "completion_tokens": 1299 }, "timestamp": "2026-03-31T02:06:00.288Z", "answer": 70800 }, { "...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "ONE_FACTORIAL_0", "status": "ok" }, { "lemma": "STARS_BA...
{ "lo": -5.37, "mid": 0.23, "hi": 5.22 }
d849c5
alg_qf_psd_orbit_v1_601307018_3193
Let $M$ be the number of ordered pairs $(a, b)$ of positive integers with $1 \le a, b \le 20$ such that $25b^2 - 18ab + 10a^2 \le 4885$. Let $S$ be the set of ordered pairs $(a2, b2)$ with $1 \le a2, b2 \le 40$ satisfying $-18a2b2 + 25b2^2 + 10a2^2 \le 3904$, and let $N = |S|$. Let $Q$ be the number of ordered pairs $(...
5
graphs = [ Graph( let={ "_m": Const(3904), "_n": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var("a"), Var("b")]), condition=And(Geq(Var("a"), Const(1)), Leq(Var("a"), Const(20)), Geq(Var("b"), Const(1)), Leq(Var("b"), Const(20)), Leq(Sum(Mul(Const(25), Pow(Var("b"), Const(2)))...
ALG
null
COUNT
sympy
LIN_FORM
[ "QF_PSD_COUNT_LEQ/QF_PSD_COUNT_LEQ" ]
cbd80a
alg_qf_psd_orbit_v1
null
7
0
[ "LIN_FORM", "QF_PSD_COUNT_LEQ" ]
2
2.832
2026-03-10T03:45:14.853505Z
{ "verified": true, "answer": 5, "timestamp": "2026-03-10T03:45:17.685181Z" }
5be3ab
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 1, "correct": { "strict": false, "boxed": false, "relaxed": true }, "usage": { "prompt_tokens": 292, "completion_tokens": 32768 }, "timestamp": "2026-03-29T07:49:27.631Z", "answer": null }, { ...
1
[ { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" }, { "lemma": "QF_PSD_COUNT_LEQ", "status": "ok" } ]
{ "lo": 1.23, "mid": 4.27, "hi": 6.7 }
03e703
antilemma_product_of_sums_v1_124444284_248
Let $S_1$ be the sum of $k$ over all ordered pairs $(k, j)$ where $k$ ranges from 1 to 8 and $j$ ranges from 1 to 5. Let $d_{\text{min}}$ be the smallest divisor of 1001 that is at least 2. Define $S_2$ to be the sum of all nonnegative integers $j \leq 7$ for which the binomial coefficient $\binom{d_{\text{min}}}{j}$ i...
10,013
graphs = [ Graph( let={ "_n": Const(1001), "S1": 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(5)))), expr...
NT
null
COMPUTE
sympy
MIN_PRIME_FACTOR
[ "MIN_PRIME_FACTOR", "PRODUCT_OF_SUMS" ]
4fd6ac
antilemma_product_of_sums_v1
null
6
0
[ "MIN_PRIME_FACTOR", "PRODUCT_OF_SUMS" ]
2
0.003
2026-02-08T03:06:13.473419Z
{ "verified": true, "answer": 10013, "timestamp": "2026-02-08T03:06:13.476089Z" }
9f1023
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 299, "completion_tokens": 1104 }, "timestamp": "2026-02-09T15:07:34.154Z", "answer": 10013 }, { "...
1
[ { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "MIN_PRIME_FACTOR", "status": "ok" }, { "lemma": "MOD_SUB", "status": "no" }, { "lemma": "PRODUCT_OF_SUMS", "statu...
{ "lo": -6.51, "mid": -0.32, "hi": 5.36 }
a55beb
geo_count_lattice_rect_v1_1978505735_6306
Compute the number of lattice points in the rectangle $[0, 128] \times [0, 302]$, including the boundary.
39,087
graphs = [ Graph( let={ "a": Const(128), "b": Const(302), "result": LatticePointsRect(a=Ref(name='a'), b=Ref(name='b')), }, goal=Ref("result"), ) ]
GEOM
null
COUNT
sympy
[]
geo_count_lattice_rect_v1
null
2
0
null
null
0.001
2026-02-08T19:33:09.674219Z
{ "verified": true, "answer": 39087, "timestamp": "2026-02-08T19:33:09.674922Z" }
4ad48e
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 76, "completion_tokens": 497 }, "timestamp": "2026-02-25T01:26:34.226Z", "answer": 39087 }, { ...
1
[]
{ "lo": -8.48, "mid": -5.37, "hi": -3.03 }
67de60
nt_count_divisible_and_v1_1915831931_545
Let $d_1 = 6$. Define $p$ to be a positive integer such that there exists a positive integer $q$ with $p \cdot q = 5880$, $\gcd(p, q) = 1$, and $p < q$. Let $d_2$ be the number of such integers $p$. Let $U = 53688$. Define $n$ to be an integer satisfying $1 \leq n \leq U$, $d_1$ divides $n$, and $d_2$ divides $n$. Let ...
13,347
graphs = [ Graph( let={ "_n": Const(60358), "upper": Const(53688), "d1": Const(6), "d2": 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')),...
NT
null
COUNT
sympy
COPRIME_PAIRS
[ "COPRIME_PAIRS" ]
2bb3aa
nt_count_divisible_and_v1
null
6
0
[ "COPRIME_PAIRS" ]
1
3.31
2026-02-08T15:31:03.760413Z
{ "verified": true, "answer": 13347, "timestamp": "2026-02-08T15:31:07.070635Z" }
af5897
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 193, "completion_tokens": 2317 }, "timestamp": "2026-02-16T08:50:24.460Z", "answer": 13347 }, ...
1
[ { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "K15", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" }, { "lemma": "V5", "s...
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
dd2207
nt_min_coprime_above_v1_655260480_4728
Let $r$ be the sum of the solutions to the equation $x^2 - 493x - 1988 = 0$. Find the smallest integer $n$ such that $40000 < n \leq 40503$ and $\gcd(n, r) = 1$. Let $Q$ be the Bell number corresponding to the remainder when the absolute value of this $n$ is divided by $11$. Compute $Q$.
203
graphs = [ Graph( let={ "start": Const(40000), "upper": Const(40503), "modulus": SumOverSet(set=SolutionsSet(var=Var("x"), condition=Eq(Sum(Pow(Var("x"), Const(2)), Mul(Const(-493), Var("x")), Const(-1988)), Const(0)))), "result": MinOverSet(set=SolutionsSet(v...
NT
COMB
EXTREMUM
sympy
VIETA_SUM
[ "VIETA_SUM" ]
b33a7a
nt_min_coprime_above_v1
null
5
0
[ "VIETA_SUM" ]
1
0.073
2026-02-08T18:05:06.561116Z
{ "verified": true, "answer": 203, "timestamp": "2026-02-08T18:05:06.633781Z" }
72798a
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 138, "completion_tokens": 1210 }, "timestamp": "2026-02-18T13:57:20.083Z", "answer": 203 }, { ...
1
[ { "lemma": "K14", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "L3b", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "V8", "status": "no" }, { "lemma": "VIETA_SUM", "status": "ok" } ]
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
9c4cb0
nt_count_divisors_in_range_v1_677425708_2693
Let $n = 166320$. Let $a$ be the smallest integer $d \geq 2$ that divides $55190041$. Let $b = 11880$. Determine the number of positive divisors $d$ of $n$ such that $a \leq d \leq b$.
133
graphs = [ Graph( let={ "n": Const(166320), "a": MinOverSet(set=SolutionsSet(var=Var("d"), condition=And(Geq(Var("d"), Const(2)), Divides(divisor=Var("d"), dividend=Const(55190041))))), "b": Const(11880), "result": CountOverSet(set=SolutionsSet(var=Var("d"), c...
NT
null
COUNT
sympy
MIN_PRIME_FACTOR
[ "MIN_PRIME_FACTOR" ]
bc3776
nt_count_divisors_in_range_v1
null
4
0
[ "MIN_PRIME_FACTOR" ]
1
0.106
2026-02-08T05:12:07.849965Z
{ "verified": true, "answer": 133, "timestamp": "2026-02-08T05:12:07.955733Z" }
2a0397
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 176, "completion_tokens": 6240 }, "timestamp": "2026-02-11T23:04:41.956Z", "answer": 133 }, { "i...
1
[ { "lemma": "K13", "status": "no" }, { "lemma": "K15", "status": "no" }, { "lemma": "L3b", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MIN_PRIME_FACTOR", "status": "ok" }, { "lemma": "MOD_MUL", "status": "no" } ]
{ "lo": -3.42, "mid": 1.86, "hi": 7.55 }
afe561
diophantine_fbi2_count_v1_865884756_3605
Let $n$ be the minimum value of $x + y$ over all ordered pairs $(x, y)$ of positive integers such that $xy = 6760000$. Let $k = 60$ and $m = 6$. Let $T$ be the set of all positive integers $k_1$ such that $1 \leq k_1 \leq n$ and $240$ divides $k_1$, and let $t$ be the number of elements in $T$. Determine the number of ...
2
graphs = [ Graph( let={ "_c": Const(65), "_m": Const(6), "_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(6760000)))), e...
NT
null
COUNT
sympy
B3
[ "B3/C2", "C2/C2" ]
67f475
diophantine_fbi2_count_v1
null
7
0
[ "B3", "C2" ]
2
0.011
2026-02-08T17:30:57.250919Z
{ "verified": true, "answer": 2, "timestamp": "2026-02-08T17:30:57.261619Z" }
181cc3
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 193, "completion_tokens": 1671 }, "timestamp": "2026-02-18T03:33:17.341Z", "answer": 2 }, { ...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "C2", "status": "ok" }, { "lemma": "K14", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "L3c", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" }, { "lemma": "MA...
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
dcaad5
comb_bell_compute_v1_151522320_1975
Let $k_{\text{max}}$ be the largest integer $k$ such that $2^k \leq 39341198735$. Let $n$ be the number of positive integers $n'$ such that $1 \leq n' \leq 72$, $6$ divides $n'$, and $\gcd(n', k_{\text{max}}) = 1$. Compute the $n$th Bell number, which counts the number of partitions of a set of size $n$. Find the value...
21,147
graphs = [ Graph( let={ "_m": Const(6), "_n": MaxOverSet(set=SolutionsSet(var=Var("k"), condition=Leq(Pow(Const(2), Var("k")), Const(39341198735)))), "n": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), Const(72)), Divides(...
NT
COMB
COMPUTE
sympy
MAX_VAL
[ "MAX_VAL/C5" ]
33e5b1
comb_bell_compute_v1
null
7
0
[ "C5", "MAX_VAL" ]
2
0.002
2026-02-08T04:29:46.218674Z
{ "verified": true, "answer": 21147, "timestamp": "2026-02-08T04:29:46.220831Z" }
6f546f
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 225, "completion_tokens": 1026 }, "timestamp": "2026-02-10T16:49:38.015Z", "answer": 21147 }, { ...
1
[ { "lemma": "C5", "status": "ok_later" }, { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "K15", "status": "no" }, { "lemma": "K16", "status": "same_pattern_wrong" }, { "lemma": "K5", "status": "no" }, { "lemma": "MAX_VAL", "status": "ok" ...
{ "lo": -3.42, "mid": 1.86, "hi": 7.55 }
f0c2c8
antilemma_cartesian_v1_655260480_4103
Let $x$ be the number of ordered pairs $(a,b)$ such that $1 \leq a \leq 21$ and $1 \leq b \leq 42$. Let $m = x + 2$. The Fibonacci entry point modulo $m$ is defined as the smallest positive integer $k$ such that the $k$th Fibonacci number $F_k$ is divisible by $m$. Find the value of this entry point.
126
graphs = [ Graph( let={ "x": CountOverSet(set=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(21)), right=IntegerRange(start=Const(1), end=Const(42)))), "Q": FibonacciEntryPoint(modulus=Sum(Abs(arg=Ref(name='x')), Const(value=2))), }, goal=Ref("Q"), )...
COMB
GEOM
COMPUTE
sympy
COUNT_CARTESIAN
[ "COUNT_CARTESIAN" ]
409d7d
antilemma_cartesian_v1
null
2
0
[ "COUNT_CARTESIAN" ]
1
0.008
2026-02-08T17:43:35.224343Z
{ "verified": true, "answer": 126, "timestamp": "2026-02-08T17:43:35.232686Z" }
31cfd3
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 138, "completion_tokens": 1395 }, "timestamp": "2026-02-18T07:43:46.370Z", "answer": 126 }, { ...
1
[ { "lemma": "COUNT_CARTESIAN", "status": "ok" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", ...
{ "lo": -2.38, "mid": 1.74, "hi": 6.59 }
128a3c
comb_sum_binomial_row_v1_1218484723_7659
Let $n$ be the number of ordered pairs $(a, b)$ of positive integers with $1 \le a, b \le 15$ such that $17a^4 + 68a^3b + 102a^2b^2 + 68ab^3 + 17b^4 = 1114112$. Compute $2^n$.
32,768
graphs = [ Graph( let={ "_n": Const(17), "n": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var("a"), Var("b")]), condition=And(Geq(Var("a"), Const(1)), Leq(Var("a"), Const(15)), Geq(Var("b"), Const(1)), Leq(Var("b"), Const(15)), Eq(Sum(Mul(Ref("_n"), Pow(Var("b"), Const(4))), Mu...
COMB
null
SUM
sympy
POLY_ORBIT_HENSEL
[ "POLY4_COUNT" ]
861d91
comb_sum_binomial_row_v1
null
3
0
[ "POLY4_COUNT", "POLY_ORBIT_HENSEL" ]
2
0.357
2026-02-25T09:07:11.779829Z
{ "verified": true, "answer": 32768, "timestamp": "2026-02-25T09:07:12.136915Z" }
559d72
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 188, "completion_tokens": 643 }, "timestamp": "2026-03-30T05:45:59.785Z", "answer": 32768 }, { "i...
1
[ { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "POLY4_COUNT", "status": "ok" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", "status": "no" }, { "lemma": "V8_SUM", "status": ...
{ "lo": -6.2, "mid": -2.87, "hi": 0.44 }
988585
alg_sum_powers_v1_1218484723_2035
Let $M$ be the largest positive divisor of $25060027$ that is at most $5003$. Let $P = \min\{ x + y : x, y > 0,\ xy = 772641 \}$. Compute $\left( \sum_{k=1}^{P} k^3 \right) \bmod M$.
4,726
graphs = [ Graph( let={ "_m": Const(3), "_n": MaxOverSet(set=SolutionsSet(var=Var("d"), condition=And(Geq(Var("d"), Const(1)), Leq(Var("d"), Const(5003)), Divides(divisor=Var("d"), dividend=Const(25060027))))), "result": Mod(value=Summation(var="k", start=Const(1), end=Mi...
NT
null
COMPUTE
sympy
MAX_DIVISOR
[ "MAX_DIVISOR/B3" ]
51e324
alg_sum_powers_v1
null
5
0
[ "B3", "MAX_DIVISOR" ]
2
0.07
2026-02-25T03:43:49.386095Z
{ "verified": true, "answer": 4726, "timestamp": "2026-02-25T03:43:49.455878Z" }
c11f7f
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 184, "completion_tokens": 32768 }, "timestamp": "2026-03-29T02:41:37.552Z", "answer": null }, { ...
1
[ { "lemma": "B3", "status": "ok_later" }, { "lemma": "K13", "status": "no" }, { "lemma": "L3b", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "MAX_DIVISOR", "status": "ok" }, { "lemma": "V3", "status": "no" }, { "lemm...
{ "lo": 1.14, "mid": 4.13, "hi": 6.32 }
4c8e2c
modular_min_linear_v1_1742523217_234
Let $a = 81921$, $b = 61383$, and $m = 86499$. Let $S$ be the set of all positive integers $p$ for which there exists a positive integer $q$ such that $pq = 72$, $\gcd(p, q) = 1$, and $p < q$. Let $\phi$ denote Euler's totient function evaluated at the number of elements in $S$. Determine the value of the smallest inte...
988
graphs = [ Graph( let={ "a": Const(81921), "b": Const(61383), "m": Const(86499), "result": MinOverSet(set=SolutionsSet(var=Var("x"), condition=And(Geq(Var("x"), EulerPhi(n=CountOverSet(set=SolutionsSet(var=Var("p"), condition=And(IsPositive(arg=Var(name='p')),...
NT
null
EXTREMUM
sympy
COPRIME_PAIRS
[ "COPRIME_PAIRS/ONE_PHI_2" ]
761f00
modular_min_linear_v1
null
6
0
[ "COPRIME_PAIRS", "ONE_PHI_2" ]
2
6.73
2026-02-08T02:56:26.069749Z
{ "verified": true, "answer": 988, "timestamp": "2026-02-08T02:56:32.799445Z" }
97c6c7
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 248, "completion_tokens": 4029 }, "timestamp": "2026-02-09T15:18:09.279Z", "answer": 988 }, { "id...
1
[ { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "L3c", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "ONE_PHI_2", ...
{ "lo": -6.51, "mid": -0.31, "hi": 5.43 }
3564b1
nt_min_crt_v1_865884756_2389
Let $a = 0$, $b = 10$, $m = 5$, and $k = 11$. Let $u = \sum_{k_2=1}^4 k_2$ and define $s = \sum_{k_1=1}^u k_1$. Let $r$ be the smallest positive integer $n$ such that $n \leq s$, $n \equiv a \pmod{m}$, and $n \equiv b \pmod{k}$. Compute the remainder when $91661 \cdot r$ is divided by $87980$. Find the value of this e...
36,810
graphs = [ Graph( let={ "_n": Const(87980), "m": Const(5), "k": Const(11), "a": Const(0), "b": Const(10), "upper": Summation(var="k1", start=Const(1), end=Summation(var="k2", start=Const(1), end=Const(4), expr=Var("k2")), expr=Var("k1")...
NT
null
EXTREMUM
sympy
SUM_ARITHMETIC
[ "SUM_ARITHMETIC/SUM_ARITHMETIC" ]
2a57af
nt_min_crt_v1
null
4
0
[ "SUM_ARITHMETIC" ]
1
0.008
2026-02-08T16:43:41.327248Z
{ "verified": true, "answer": 36810, "timestamp": "2026-02-08T16:43:41.334831Z" }
4673e3
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 176, "completion_tokens": 650 }, "timestamp": "2026-02-17T11:00:58.839Z", "answer": 36810 }, {...
1
[ { "lemma": "K17", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MOD_ADD", "status": "no" }, { "lemma": "SUM_ARITHMETIC", "status": "ok" } ]
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
3dc37f
antilemma_cartesian_v1_2051736721_1816
Let $x$ be the number of ordered pairs $(a, b)$ such that $1 \leq a \leq 36$ and $1 \leq b \leq 37$.\\ Let $y$ be the number of ordered pairs $(c, d)$ such that $1 \leq c \leq 7$ and $1 \leq d \leq 11$.\\ Compute the remainder when $y - x$ is divided by $74245$.
72,990
graphs = [ Graph( let={ "_n": Const(74245), "x": CountOverSet(set=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(36)), right=IntegerRange(start=Const(1), end=Const(37)))), "Q": Mod(value=Sub(CountOverSet(set=CartesianProduct(left=IntegerRange(start=Const(1),...
COMB
GEOM
COMPUTE
sympy
COUNT_CARTESIAN
[ "COUNT_CARTESIAN", "COUNT_CARTESIAN" ]
f9c395
antilemma_cartesian_v1
negation_mod
2
0
[ "COUNT_CARTESIAN" ]
1
0.002
2026-02-08T16:13:10.037915Z
{ "verified": true, "answer": 72990, "timestamp": "2026-02-08T16:13:10.039894Z" }
cc3bff
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 214, "completion_tokens": 507 }, "timestamp": "2026-02-24T20:24:52.645Z", "answer": 72990 }, { "...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_CARTESIAN", "status": "ok" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8_SUM", "status":...
{ "lo": -6.4, "mid": -4.13, "hi": -2.01 }
3ec33b
nt_count_divisors_in_range_v1_2051736721_557
Let $n = 45360$. Let $A$ be the set of all divisors $d$ of $n$ such that $40 \le d \le 2169$, where $2169$ is the number of ordered pairs $(i,j)$ with $1 \le i \le 9$ and $1 \le j \le 241$. Let $r = |A|$. Compute $\sum_{k=1}^{r} \tau(k)$, where $\tau(k)$ denotes the number of positive divisors of $k$.
263
graphs = [ Graph( let={ "n": Const(45360), "a": Const(40), "b": CountOverSet(set=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(9)), right=IntegerRange(start=Const(1), end=Const(241)))), "result": CountOverSet(set=SolutionsSet(var=Var("d"), condi...
NT
null
COUNT
sympy
COUNT_CARTESIAN
[ "COUNT_CARTESIAN" ]
409d7d
nt_count_divisors_in_range_v1
null
4
0
[ "COUNT_CARTESIAN" ]
1
0.041
2026-02-08T15:31:35.609726Z
{ "verified": true, "answer": 263, "timestamp": "2026-02-08T15:31:35.651193Z" }
3cea0b
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 159, "completion_tokens": 3659 }, "timestamp": "2026-02-16T09:03:32.639Z", "answer": 263 }, { ...
1
[ { "lemma": "COUNT_CARTESIAN", "status": "ok" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "K13", "status": "no" }, { "lemma": "K15", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "V8", "status": "no" } ...
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
1a1d4f
diophantine_product_count_v1_717093673_2879
Let $k = 240$ and let $u$ be the number of ordered pairs $(i, j)$ of positive integers with $1 \leq i \leq 15$ and $1 \leq j \leq 16$ such that $\gcd(i, j) = 1$. Determine the number of positive integers $x$ such that $1 \leq x \leq u$, $x$ divides $k$, and $\frac{k}{x} \leq u$.
18
graphs = [ Graph( let={ "k": Const(240), "upper": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var("i"), Var("j")]), condition=Eq(GCD(a=Var("i"), b=Var("j")), Const(1)), domain=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(15)), right=IntegerRange(start=Const(1), ...
NT
null
COUNT
sympy
COUNT_COPRIME_GRID
[ "COUNT_COPRIME_GRID" ]
20ec03
diophantine_product_count_v1
null
5
0
[ "COUNT_COPRIME_GRID" ]
1
0.279
2026-02-08T17:15:29.090981Z
{ "verified": true, "answer": 18, "timestamp": "2026-02-08T17:15:29.370018Z" }
8ef8a5
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 145, "completion_tokens": 2000 }, "timestamp": "2026-02-17T23:05:35.522Z", "answer": 18 }, { ...
1
[ { "lemma": "COUNT_COPRIME_GRID", "status": "ok" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "L3b", "status": "no" }, { "lemma": "V8_SUM", "status": "no" }, { "lemma": "VAL_SUM_EQ", "status":...
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
0d912d
nt_count_coprime_v1_677425708_2573
Let $k$ be the smallest integer $d \geq 2$ that divides 2021. Let $\text{upper} = 21609$. Compute the number of positive integers $n$ such that $1 \leq n \leq \text{upper}$ and $\gcd(n, k) = 1$.
21,107
graphs = [ Graph( let={ "_n": Const(2), "upper": Const(21609), "k": MinOverSet(set=SolutionsSet(var=Var("d"), condition=And(Geq(Var("d"), Ref("_n")), Divides(divisor=Var("d"), dividend=Const(2021))))), "result": CountOverSet(set=SolutionsSet(var=Var("n"), cond...
NT
null
COUNT
sympy
MIN_PRIME_FACTOR
[ "MIN_PRIME_FACTOR" ]
bc3776
nt_count_coprime_v1
null
4
0
[ "MIN_PRIME_FACTOR" ]
1
1.922
2026-02-08T05:07:35.225892Z
{ "verified": true, "answer": 21107, "timestamp": "2026-02-08T05:07:37.148200Z" }
e6c91c
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 180, "completion_tokens": 831 }, "timestamp": "2026-02-11T22:57:43.589Z", "answer": 21107 }, { "...
1
[ { "lemma": "K18", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "MIN_PRIME_FACTOR", "status": "ok" }, { "lemma": "MOD_ADD", "status": "no" }, { "lemma": "V8", "status": "no" }, { "lemma": "V8_SUM", "status": "no" } ]
{ "lo": -7.01, "mid": -4.68, "hi": -2.15 }
c28886
antilemma_k3_v1_1915831931_1153
Let $n = 86687$. Define $x$ to be the sum of Euler's totient function $\phi(d)$ over all positive divisors $d$ of $n$. Let $c = 73396$. Compute the remainder when $c \cdot x$ is divided by $84847$.
57,063
graphs = [ Graph( let={ "x": SumOverDivisors(n=Const(value=86687), var='d', expr=EulerPhi(n=Var(name='d'))), "_c": Const(73396), "Q": Mod(value=Mul(Ref("_c"), Ref("x")), modulus=Const(84847)), }, goal=Ref("Q"), ) ]
NT
COMB
COMPUTE
sympy
K3
[ "K3" ]
54c41e
antilemma_k3_v1
null
3
0
[ "K3" ]
1
0.001
2026-02-08T15:54:51.411405Z
{ "verified": true, "answer": 57063, "timestamp": "2026-02-08T15:54:51.412169Z" }
fdaaab
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 108, "completion_tokens": 1135 }, "timestamp": "2026-02-16T16:33:11.118Z", "answer": 57063 }, ...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "K14", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "K3", "status": "ok" }, { "lemma": "LTE_SUM", "status": "no" } ]
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
7b2b65
comb_factorial_compute_v1_677425708_3299
Let $n$ be the number of positive integers $p$ for which there exists a positive integer $q$ such that $p \cdot q = 205800$, $\gcd(p, q) = 1$, and $p < q$. Let $c = 49695$. Compute the remainder when $c \cdot n!$ is divided by 58288.
52,400
graphs = [ Graph( let={ "_n": Const(58288), "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=205800)), Eq(left=GCD(a=Var(name='p'), b=Var(...
NT
null
COMPUTE
sympy
COPRIME_PAIRS
[ "COPRIME_PAIRS" ]
2bb3aa
comb_factorial_compute_v1
null
6
0
[ "COPRIME_PAIRS" ]
1
0.006
2026-02-08T05:38:19.203604Z
{ "verified": true, "answer": 52400, "timestamp": "2026-02-08T05:38:19.209790Z" }
f0679a
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 124, "completion_tokens": 4925 }, "timestamp": "2026-02-12T12:16:02.374Z", "answer": 52400 }, ...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "DS2", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "L3c", "status": "no" }, { "lemma": "V8_SUM", "status": "no" } ]
{ "lo": -3.44, "mid": 1.34, "hi": 6.56 }
5d7fa5
diophantine_fbi2_min_v1_1520064083_5999
Let $A$ be the set of all prime numbers $n$ such that $2 \leq n \leq 501$. Let $u$ be the number of elements $n$ in $A$ for which the sum of the decimal digits of $n$ is odd. Determine the smallest positive integer $d$ such that $5 \leq d \leq u$, $d$ divides $240$, and $\frac{240}{d} \geq 4$. Let this value be $r$. Co...
11
graphs = [ Graph( let={ "_m": Const(2), "_n": Const(4), "k": Const(240), "upper": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), MaxOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Ref("_m")),...
NT
null
EXTREMUM
sympy
MAX_PRIME_BELOW
[ "MAX_PRIME_BELOW/L3B" ]
5f10c3
diophantine_fbi2_min_v1
null
6
0
[ "L3B", "MAX_PRIME_BELOW" ]
2
0.012
2026-02-08T07:46:13.035289Z
{ "verified": true, "answer": 11, "timestamp": "2026-02-08T07:46:13.047562Z" }
05cf47
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 212, "completion_tokens": 2183 }, "timestamp": "2026-02-13T12:16:18.010Z", "answer": 11 }, { ...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "L3B", "status": "ok_later" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok" }, { "lemma": "MOD_ADD", "status": "no...
{ "lo": -5.55, "mid": -3.01, "hi": 0.32 }
5497bc
alg_poly4_sum_v1_601307018_9995
Let $p_{\max}$ be the largest prime $n$ such that $2 \le n \le 452$. Compute the remainder when $$\sum_{a=1}^{p_{\max}} \sum_{b=1}^{449} (82a^4 - 212a^3b + 222a^2b^2 - 92ab^3 + 17b^4)$$ is divided by $86303$.
61,155
graphs = [ Graph( let={ "_n": Const(86303), "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"), MaxOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(2)), L...
NT
null
COMPUTE
sympy
MAX_PRIME_BELOW
[ "MAX_PRIME_BELOW" ]
dc3ad3
alg_poly4_sum_v1
null
5
0
[ "MAX_PRIME_BELOW" ]
1
0.84
2026-03-10T10:26:16.853785Z
{ "verified": true, "answer": 61155, "timestamp": "2026-03-10T10:26:17.693824Z" }
74d2f7
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 202, "completion_tokens": 9885 }, "timestamp": "2026-04-19T12:46:05.341Z", "answer": 61155 }, { ...
1
[ { "lemma": "K5", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" }, { "lemma": "V3", "status": "no" }, { "lemma": "VAL_SUM_EQ", "status": "...
{ "lo": -5.34, "mid": 1.1, "hi": 7.53 }
48b7d6
v1_endings_v1_1874849503_361
Let $n = 31966$ and $p = 5$. Let $v_p(n!)$ denote the largest integer $k$ such that $p^k$ divides $n!$. Compute the value of $v_p(n!) - \left\lfloor \log_p n \right\rfloor$.
7,983
graphs = [ Graph( let={ "n_val": Const(31966), "p_val": Const(5), "n_fact": Factorial(Ref("n_val")), "vp_fact": MaxKDivides(target=Ref("n_fact"), base=Ref("p_val")), "log_p_n": Floor(Log(left=Ref(name='n_val'), right=Ref(name='p_val'))), ...
NT
null
COMPUTE
sympy
V1
[ "V1" ]
dae96f
v1_endings_v1
null
3
null
[ "V1" ]
1
0.001
2026-02-08T12:58:04.329697Z
{ "verified": true, "answer": 7983, "timestamp": "2026-02-08T12:58:04.330438Z" }
bc2d1f
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 176, "completion_tokens": 832 }, "timestamp": "2026-02-09T16:10:02.207Z", "answer": 7983 }, { "id...
1
[ { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "K5", "status": "same_pattern_wrong" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" }, { "lemma": "V1", "status": "ok" }, { "lemma": "V5", "status":...
{ "lo": -6.5, "mid": -0.2, "hi": 6.11 }
e7ab02
comb_binomial_compute_v1_865884756_6481
Let $n = 13$. Let $k$ be the smallest integer $d$ such that $d \geq 2$ and $d$ divides $2695$. Compute $\binom{n}{k}$.
1,287
graphs = [ Graph( let={ "_n": Const(2), "n": Const(13), "k": MinOverSet(set=SolutionsSet(var=Var("d"), condition=And(Geq(Var("d"), Ref("_n")), Divides(divisor=Var("d"), dividend=Const(2695))))), "result": Binom(n=Ref("n"), k=Ref("k")), }, goal=...
NT
null
COMPUTE
sympy
MIN_PRIME_FACTOR
[ "MIN_PRIME_FACTOR" ]
bc3776
comb_binomial_compute_v1
null
3
0
[ "MIN_PRIME_FACTOR" ]
1
0.002
2026-02-08T19:13:39.056366Z
{ "verified": true, "answer": 1287, "timestamp": "2026-02-08T19:13:39.058280Z" }
75ae7f
CC BY 4.0
[ { "id": 8, "model": "mathstral", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 104, "completion_tokens": 752 }, "timestamp": "2026-02-16T18:36:20.022Z", "answer": 1287 }, { "id": 11, ...
2
[ { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "MIN_PRIME_FACTOR", "status": "ok" }, { "lemma": "MOD_MUL", "status": "no" }, { "lemma": "V7", ...
{ "lo": -10, "mid": -7.27, "hi": -4.54 }
9791f0
antilemma_k3_v1_48377204_3047
Let $m = 83841$ and $n = 27714$. Define $x$ to be the sum of $\phi(d)$ over all positive divisors $d$ of $n$, where $\phi$ is Euler's totient function. Let $S$ be the sum of all real solutions $x_1$ to the equation $x_1^2 - 7690x_1 - 605904 = 0$. Compute the remainder when $S \cdot x$ is divided by $m$.
80,679
graphs = [ Graph( let={ "_m": Const(83841), "_n": Const(27714), "x": SumOverDivisors(n=Ref(name='_n'), var='d', expr=EulerPhi(n=Var(name='d'))), "Q": Mod(value=Mul(SumOverSet(set=SolutionsSet(var=Var("x1"), condition=Eq(Sum(Pow(Var("x1"), Const(2)), Mul(Const(...
NT
COMB
COMPUTE
sympy
K13
[ "VIETA_SUM", "IDENTITY_MUL_ZERO", "K3" ]
23907a
antilemma_k3_v1
affine_mod
3
0
[ "IDENTITY_MUL_ZERO", "K13", "K3", "VIETA_SUM" ]
4
0.004
2026-02-08T17:08:48.561804Z
{ "verified": true, "answer": 80679, "timestamp": "2026-02-08T17:08:48.565427Z" }
ed4706
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 153, "completion_tokens": 2159 }, "timestamp": "2026-02-17T20:33:04.744Z", "answer": 80679 }, ...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "IDENTITY_MUL_ZERO", "status": "ok" }, { "lemma": "K13", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "K3", "status": "ok" }, { "lemma": "K5", "status": "no" }, { "lemma": "M...
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
3caba0
modular_min_linear_v1_1742523217_3246
Let $a = 19115$ and $m = 52046$. Let $b$ be the number of integers $n$ with $1 \leq n \leq 16561$ such that the sum of the decimal digits of $n$ is odd. Determine the smallest positive integer $x$ such that $1 \leq x \leq m$ and $$ a \cdot x \equiv b \pmod{m}. $$
12,999
graphs = [ Graph( let={ "_n": Const(2), "a": Const(19115), "b": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), Const(16561)), Eq(Mod(value=DigitSum(Var("n")), modulus=Ref("_n")), Const(1))))), "m": Const(52046)...
NT
null
EXTREMUM
sympy
L3B
[ "L3B" ]
cc148f
modular_min_linear_v1
null
6
0
[ "L3B" ]
1
2.097
2026-02-08T05:44:53.080736Z
{ "verified": true, "answer": 12999, "timestamp": "2026-02-08T05:44:55.177239Z" }
6aed46
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 137, "completion_tokens": 5061 }, "timestamp": "2026-02-12T13:35:22.836Z", "answer": 12999 }, ...
1
[ { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "K18", "status": "no" }, { "lemma": "L3B", "status": "ok" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, { "lemma": "POLY_PADIC_VAL_CONST", ...
{ "lo": -3.51, "mid": 1.33, "hi": 6.56 }
21a603
comb_binomial_compute_v1_1978505735_7446
Let $m$ be the maximum value of $x_1 y_1$ over all ordered pairs $(x_1, y_1)$ of positive integers such that $x_1 + y_1 = 14$. Let $n$ be the minimum value of $x + y$ over all ordered pairs $(x, y)$ of positive integers such that $xy = m$. Compute $\binom{n}{7}$.
3,432
graphs = [ Graph( let={ "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")), MaxOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var("x1"), Var(...
ALG
COMB
COMPUTE
sympy
B1
[ "B1/B3" ]
80b49d
comb_binomial_compute_v1
null
3
0
[ "B1", "B3" ]
2
0.003
2026-02-08T20:16:19.100003Z
{ "verified": true, "answer": 3432, "timestamp": "2026-02-08T20:16:19.102991Z" }
4b24ba
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 134, "completion_tokens": 1149 }, "timestamp": "2026-02-25T01:54:19.772Z", "answer": 3432 }, {...
1
[ { "lemma": "B1", "status": "ok" }, { "lemma": "B3", "status": "ok_later" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" } ]
{ "lo": -8.48, "mid": -5.37, "hi": -3.03 }
c1250a
sequence_fibonacci_compute_v1_677425708_828
Let $n$ be the largest prime number satisfying $2 \leq n \leq 23$. Compute the remainder when $73181$ times the $n$th Fibonacci number is divided by $82064$.
2,397
graphs = [ Graph( let={ "_n": Const(2), "n": MaxOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Ref("_n")), Leq(Var("n"), Const(23)), IsPrime(Var("n"))))), "result": Fibonacci(arg=Ref(name='n')), "Q": Mod(value=Mul(Const(73181), Ref("result"...
NT
null
COMPUTE
sympy
MAX_PRIME_BELOW
[ "MAX_PRIME_BELOW" ]
dc3ad3
sequence_fibonacci_compute_v1
null
3
0
[ "MAX_PRIME_BELOW" ]
1
0.001
2026-02-08T03:47:34.857115Z
{ "verified": true, "answer": 2397, "timestamp": "2026-02-08T03:47:34.858450Z" }
c3a2ec
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 159, "completion_tokens": 3564 }, "timestamp": "2026-02-09T13:30:34.899Z", "answer": 2397 }, { "i...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "DS2", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok" }, { "lemma": "MOD_ADD", "status": "no" }, { "lemma": "MOD_MUL", "status": "no" } ]
{ "lo": -3.46, "mid": 1.03, "hi": 5.49 }
a8949f
antilemma_product_of_sums_v1_168721529_715
Let $S_1$ be the set of all ordered pairs $(k,\_j)$ with $1 \leq k \leq 8$ and $1 \leq \_j \leq 4$. Let $a$ be the sum of all values of $k$ as $(k,\_j)$ ranges over $S_1$. Let $S_2$ be the set of all ordered pairs $(i,j)$ with $1 \leq i \leq 9$ and $1 \leq j \leq 5$. Let $b$ be the sum of all values of $ij$ as $(i,j)$ ...
44,027
graphs = [ Graph( let={ "x": Mul(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(4)))), expr=Var("k"))), SumOverSet(set=...
NT
null
COMPUTE
sympy
PRODUCT_OF_SUMS
[ "PRODUCT_OF_SUMS" ]
f2b2b0
antilemma_product_of_sums_v1
null
2
0
[ "PRODUCT_OF_SUMS" ]
1
0.001
2026-02-08T13:12:38.105927Z
{ "verified": true, "answer": 44027, "timestamp": "2026-02-08T13:12:38.106827Z" }
e8fc7f
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 285, "completion_tokens": 744 }, "timestamp": "2026-02-09T08:16:21.512Z", "answer": 44027 }, { "i...
1
[ { "lemma": "DS2", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, { "lemma": "PRODUCT_OF_SUMS", "status": "ok" }, { "lemma": "V3", "status": "no" }, { "lemma": "V7", "status": "no" }...
{ "lo": -6.69, "mid": -2.39, "hi": 1.83 }
bee470
diophantine_fbi2_count_v1_1248542787_729
Let $k = 360$. Compute the number of integers $d$ such that $3 \leq d \leq 107$, $d$ divides $k$, $\frac{k}{d} \geq 3$, and $$ \frac{k}{d} \leq \text{the number of integers } n \text{ with } 1 \leq n \leq 539 \text{ such that } n \equiv \left\lfloor \frac{n}{2} \right\rfloor \pmod{5}. $$ Let $c = 26549$. Find the rema...
78,462
graphs = [ Graph( let={ "k": Const(360), "result": CountOverSet(set=SolutionsSet(var=Var("d"), condition=And(Geq(Var("d"), Const(3)), Leq(Var("d"), Const(107)), Divides(divisor=Var("d"), dividend=Ref("k")), Geq(Div(Ref("k"), Var("d")), Const(3)), Leq(Div(Ref("k"), Var("d")), CountOve...
NT
null
COUNT
sympy
L3C
[ "L3C" ]
73f8b0
diophantine_fbi2_count_v1
null
5
0
[ "L3C" ]
1
0.008
2026-02-08T03:21:21.946816Z
{ "verified": true, "answer": 78462, "timestamp": "2026-02-08T03:21:21.955081Z" }
1494bf
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 256, "completion_tokens": 2990 }, "timestamp": "2026-02-09T20:26:54.437Z", "answer": 78462 }, { ...
1
[ { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "L3C", "status": "ok" }, { "lemma": "L3b", "status": "no" }, { "lemma": "VAL_SUM_EQ", "s...
{ "lo": -3.42, "mid": 1.86, "hi": 7.55 }
97b16b
comb_count_partitions_v1_1742523217_1029
Let $n$ be the number of integers $t$ with $18 \leq t \leq 116$ such that there exist positive integers $a$ and $b$ with $1 \leq a \leq 4$, $1 \leq b \leq 15$, and $t = 14a + 4b$. Let $P(n)$ denote the number of integer partitions of $n$. Compute the remainder when $27565 \cdot P(n)$ is divided by $56416$.
39,195
graphs = [ Graph( let={ "_n": Const(56416), "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=Va...
COMB
null
COUNT
sympy
LIN_FORM
[ "LIN_FORM" ]
7b2633
comb_count_partitions_v1
null
5
0
[ "LIN_FORM" ]
1
0.002
2026-02-08T03:23:49.101019Z
{ "verified": true, "answer": 39195, "timestamp": "2026-02-08T03:23:49.102597Z" }
4de946
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 220, "completion_tokens": 32373 }, "timestamp": "2026-02-23T22:15:55.843Z", "answer": 39195 }, { ...
1
[ { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": ...
{ "lo": 1.97, "mid": 4.36, "hi": 6.69 }
8343d2
geo_count_lattice_rect_v1_1918700295_3343
Compute the number of lattice points in the rectangle $[0, 225] \times [0, 186]$, including its boundary.
42,262
graphs = [ Graph( let={ "a": Const(225), "b": Const(186), "result": LatticePointsRect(a=Ref(name='a'), b=Ref(name='b')), }, goal=Ref("result"), ) ]
GEOM
null
COUNT
sympy
[]
geo_count_lattice_rect_v1
null
3
0
null
null
0
2026-02-08T08:32:42.521947Z
{ "verified": true, "answer": 42262, "timestamp": "2026-02-08T08:32:42.522325Z" }
61dd5e
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 139, "completion_tokens": 319 }, "timestamp": "2026-02-24T09:41:58.889Z", "answer": 42262 }, { "i...
1
[]
{ "lo": -3.84, "mid": -1.67, "hi": 1.32 }
f8f32a
comb_count_derangements_v1_349078426_1169
Let $n$ be the number of positive integers less than or equal to 17 whose digit sum is even. Compute the subfactorial of $n$.
14,833
graphs = [ Graph( let={ "_n": Const(17), "n": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), Ref("_n")), Eq(Mod(value=DigitSum(Var("n")), modulus=Const(2)), Const(0))))), "result": Subfactorial(arg=Ref(name='n')), ...
COMB
null
COUNT
sympy
L3B
[ "L3B" ]
cc148f
comb_count_derangements_v1
null
3
0
[ "L3B" ]
1
0.002
2026-02-08T13:27:14.762387Z
{ "verified": true, "answer": 14833, "timestamp": "2026-02-08T13:27:14.764205Z" }
372d17
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 142, "completion_tokens": 733 }, "timestamp": "2026-02-24T18:22:59.958Z", "answer": 14833 }, { "i...
1
[ { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "L3B", "status": "ok" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8_SUM", "sta...
{ "lo": -4.92, "mid": -2.91, "hi": -0.68 }
f9d17c
comb_count_derangements_v1_397696148_798
Let $n$ be the smallest integer $d$ such that $d \geq 2$ and $d$ divides $41503$. Compute the subfactorial of $n$.
1,854
graphs = [ Graph( let={ "_n": Const(2), "n": MinOverSet(set=SolutionsSet(var=Var("d"), condition=And(Geq(Var("d"), Ref("_n")), Divides(divisor=Var("d"), dividend=Const(41503))))), "result": Subfactorial(arg=Ref(name='n')), }, goal=Ref("result"), ) ]
NT
COMB
COUNT
sympy
MIN_PRIME_FACTOR
[ "MIN_PRIME_FACTOR" ]
bc3776
comb_count_derangements_v1
null
2
0
[ "MIN_PRIME_FACTOR" ]
1
0.001
2026-02-08T11:44:28.692786Z
{ "verified": true, "answer": 1854, "timestamp": "2026-02-08T11:44:28.693735Z" }
e66cba
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 85, "completion_tokens": 791 }, "timestamp": "2026-02-14T18:06:35.620Z", "answer": 1854 }, { ...
1
[ { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "MIN_PRIME_FACTOR", "status": "ok" }, { "lemma": "V3", "status": "no" }, { "lemma": "V5", "status": "no" ...
{ "lo": -5.14, "mid": 0.32, "hi": 6.51 }
df70e4
diophantine_fbi2_min_v1_784195855_7092
Let $k$ be the number of positive integers $n \leq 19$ such that the sum of the digits of $n$ is odd. Let $d$ be the smallest integer $d$ with $2 \leq d \leq 20$ such that $d$ divides $k$ and $\frac{k}{d} \geq 4$. Compute the remainder when $71922 \cdot d$ is divided by $90677$.
53,167
graphs = [ Graph( let={ "k": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), Const(19)), Eq(Mod(value=DigitSum(Var("n")), modulus=Const(2)), Const(1))))), "upper": Const(20), "result": MinOverSet(set=SolutionsSet(var=Var("d...
NT
null
EXTREMUM
sympy
L3B
[ "L3B" ]
cc148f
diophantine_fbi2_min_v1
null
4
0
[ "L3B" ]
1
0.004
2026-02-08T09:05:09.508153Z
{ "verified": true, "answer": 53167, "timestamp": "2026-02-08T09:05:09.512401Z" }
cb291d
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 139, "completion_tokens": 844 }, "timestamp": "2026-02-14T00:11:36.260Z", "answer": 53167 }, {...
1
[ { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "L3B", "status": "ok" }, { "lemma": "L3b", "status": "no" }, { "lemma": "MOD_ADD", "status": "no" }, { "lemma": "MOD_MUL", "status": "no" }, { "lemma": "V8_SUM", "status": "no" } ]
{ "lo": -5.55, "mid": -3.01, "hi": 0.32 }
155088
alg_poly3_min_v1_1218484723_2782
Let $M$ be the number of ordered pairs $(a, b)$ of positive integers with $1 \le a, b \le 30$ such that $13a^2 - 2ab + 2b^2 \le 832$. Let $Q$ be the minimum value of $$ 13552a_1^3 + 43560a_1^{k} b_1 + 13068a_1b_1^2 + 13068b_1^3 $$ over all positive integers $a_1, b_1$ with $1 \le a_1 \le M$, $1 \le b_1 \le 134$, where ...
83,248
graphs = [ Graph( let={ "_m": Const(14), "_n": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var("a"), Var("b")]), condition=And(Geq(Var("a"), Const(1)), Leq(Var("a"), Const(30)), Geq(Var("b"), Const(1)), Leq(Var("b"), Const(30)), Leq(Sum(Mul(Const(13), Pow(Var("a"), Const(2))), ...
ALG
null
COMPUTE
sympy
QF_PSD_COUNT_LEQ
[ "QF_PSD_COUNT_LEQ/QF_PSD_MIN" ]
7f6761
alg_poly3_min_v1
null
6
0
[ "QF_PSD_COUNT_LEQ", "QF_PSD_MIN" ]
2
0.046
2026-02-25T04:29:40.148757Z
{ "verified": true, "answer": 83248, "timestamp": "2026-02-25T04:29:40.195001Z" }
fe9909
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 332, "completion_tokens": 6126 }, "timestamp": "2026-03-29T06:32:14.714Z", "answer": 83248 }, { "...
1
[ { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" }, { "lemma": "QF_PSD_COUNT_LEQ", "status": "ok" }, { "lemma": "QF_PSD_MIN", "status": "ok_later" } ]
{ "lo": -4.26, "mid": -1.8, "hi": 1.26 }
a6040f
nt_count_intersection_v1_153355830_2359
Let $N = 20000$. Define $a$ to be the number of integers $t$ with $5 \leq t \leq 15$ such that there exist positive integers $a', b'$ with $1 \leq a' \leq 3$, $1 \leq b' \leq 3$, and $t = 2a' + 3b'$. Let $b = 14$. Define $S$ to be the set of all positive integers $n \leq N$ such that $a$ divides $n$ and $\gcd(n, b) = 1...
952
graphs = [ Graph( let={ "N": Const(20000), "a": 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...
NT
null
COUNT
sympy
LIN_FORM
[ "LIN_FORM" ]
7b2633
nt_count_intersection_v1
null
5
0
[ "LIN_FORM" ]
1
1.029
2026-02-08T07:04:50.294383Z
{ "verified": true, "answer": 952, "timestamp": "2026-02-08T07:04:51.323815Z" }
a70cf6
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 185, "completion_tokens": 1150 }, "timestamp": "2026-02-13T07:42:28.243Z", "answer": 1905 }, ...
1
[ { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "K14", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "V7", "status": "no" } ]
{ "lo": -3.44, "mid": 1.34, "hi": 6.56 }
6b8cbe
antilemma_k3_v1_865884756_5091
Compute the sum $$ \sum_{d \mid 23481} \phi(d), $$ where $\phi$ denotes Euler's totient function.
23,481
graphs = [ Graph( let={ "x": SumOverDivisors(n=Const(value=23481), var='d', expr=EulerPhi(n=Var(name='d'))), }, goal=Ref("x"), ) ]
NT
COMB
COMPUTE
sympy
K3
[ "K3" ]
54c41e
antilemma_k3_v1
null
3
0
[ "K3" ]
1
0
2026-02-08T18:22:40.204652Z
{ "verified": true, "answer": 23481, "timestamp": "2026-02-08T18:22:40.204994Z" }
b8460f
CC BY 4.0
[ { "id": 8, "model": "mathstral", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 95, "completion_tokens": 319 }, "timestamp": "2026-02-16T12:18:47.703Z", "answer": 23481 }, { "id": 11, ...
2
[ { "lemma": "C2", "status": "no" }, { "lemma": "K3", "status": "ok" }, { "lemma": "L3c", "status": "no" }, { "lemma": "MOD_SUB", "status": "no" }, { "lemma": "V8", "status": "no" }, { "lemma": "V8_SUM", "status": "no" } ]
{ "lo": -10, "mid": -7.27, "hi": -4.54 }
96f3cf
alg_poly4_sum_v1_1218484723_731
Let $P = \max\{ n : n \geq 2, n \leq 339, n \text{ is prime} \}$. Find the remainder when $\sum_{a=1}^{198} \sum_{b=1}^{198} \left( P a^4 + 296a^3b + 600a^2b^2 + 32ab^3 + 32b^4 \right)$ is divided by $54704$.
21,906
graphs = [ Graph( let={ "_n": Const(339), "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(198)), Geq(Var("b"), Const(1)), Leq(Var("b"), Const(198)))), expr=Sum(Mul(MaxO...
NT
null
COMPUTE
sympy
MAX_PRIME_BELOW
[ "MAX_PRIME_BELOW" ]
dc3ad3
alg_poly4_sum_v1
null
5
0
[ "MAX_PRIME_BELOW" ]
1
0.091
2026-02-25T02:28:02.135059Z
{ "verified": true, "answer": 21906, "timestamp": "2026-02-25T02:28:02.225857Z" }
42c86a
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 211, "completion_tokens": 32768 }, "timestamp": "2026-03-10T01:21:55.015Z", "answer": null }, { ...
1
[ { "lemma": "DS2", "status": "no" }, { "lemma": "K15", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok" }, { "lemma": "MAX_VAL", "status": "no" }, { "lemma": "VAL_SUM_EQ", "status": "no" } ]
{ "lo": 3.81, "mid": 5.7, "hi": 7.82 }
db4a65
comb_binomial_compute_v1_601307018_2852
Let $n$ be the number of integers $v$ in the range $72 \leq v \leq 1800$ for which there exist integers $a, b \in \{1, 2, 3, 4, 5\}$ such that $32a \cdot b + 32b^2 + 8a^2 = v$. Let $M = \binom{n}{6}$. Find the remainder when $44121M$ is divided by $70880$.
11,796
graphs = [ Graph( let={ "_n": Const(70880), "n": CountOverSet(set=SolutionsSet(var=Var("v"), condition=And(Geq(Var("v"), Const(72)), Leq(Var("v"), Const(1800)), Exists(var=Tuple(elements=[Var(name='a'), Var(name='b')]), condition=And(Geq(left=Var(name='a'), right=Const(value=1)), Leq...
COMB
null
COMPUTE
sympy
QF_PSD_DISTINCT
[ "QF_PSD_DISTINCT" ]
a8f9cb
comb_binomial_compute_v1
null
4
0
[ "QF_PSD_DISTINCT" ]
1
0.004
2026-03-10T03:28:56.789124Z
{ "verified": true, "answer": 11796, "timestamp": "2026-03-10T03:28:56.792903Z" }
0c03cd
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 219, "completion_tokens": 2563 }, "timestamp": "2026-03-29T06:40:59.595Z", "answer": 11796 }, { "...
1
[ { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "QF_PSD_DISTINCT", "status": "ok" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", ...
{ "lo": -2.46, "mid": 1.23, "hi": 4.93 }
f4fb01
algebra_quadratic_discriminant_v1_1520064083_8423
Let $a = -9$ and $c = 10$. Let $b$ be the number of nonnegative integers $j$ such that $0 \le j \le 321$ and $\binom{k}{j}$ is odd, where $k$ is the number of prime numbers $n$ satisfying $2 \le n \le 2131$. Compute $b^2 - 4ac$.
424
graphs = [ Graph( let={ "_n": Const(2), "a": Const(-9), "b": CountOverSet(set=SolutionsSet(var=Var("j"), condition=And(Geq(Var("j"), Const(0)), Leq(Var("j"), Const(321)), Eq(Mod(value=Binom(n=CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(2)...
NT
null
COMPUTE
sympy
COUNT_PRIMES
[ "COUNT_PRIMES/V8" ]
e03b9d
algebra_quadratic_discriminant_v1
null
6
0
[ "COUNT_PRIMES", "V8" ]
2
0.002
2026-02-08T10:10:33.404048Z
{ "verified": true, "answer": 424, "timestamp": "2026-02-08T10:10:33.406329Z" }
394345
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 132, "completion_tokens": 1705 }, "timestamp": "2026-02-14T06:41:34.499Z", "answer": 424 }, { ...
1
[ { "lemma": "COUNT_PRIMES", "status": "ok" }, { "lemma": "K13", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, { "lemma": "V1", "status": "no" }, { "lemma": "V5", "status": "no" }, { "...
{ "lo": -5.14, "mid": 0.32, "hi": 6.51 }
a1bfa6
comb_count_partitions_v1_1874849503_105
Let $n$ be the number of integers $t$ such that $12 \leq t \leq 75$ and there exist positive integers $a$, $b$ with $1 \leq a \leq 8$, $1 \leq b \leq 5$, and $t = 5a + 7b$. Let $r$ be the number of integer partitions of $n$. Find the remainder when $4 - r$ is divided by $76572$.
39,238
graphs = [ Graph( let={ "_n": Const(76572), "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=8)), Geq(left=Va...
COMB
null
COUNT
sympy
LIN_FORM
[ "LIN_FORM" ]
7b2633
comb_count_partitions_v1
null
7
0
[ "LIN_FORM" ]
1
0.001
2026-02-08T12:48:43.572524Z
{ "verified": true, "answer": 39238, "timestamp": "2026-02-08T12:48:43.573870Z" }
5282d2
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 218, "completion_tokens": 1484 }, "timestamp": "2026-02-09T13:51:07.258Z", "answer": 39238 }, { "...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8_SUM", "status": "no" ...
{ "lo": 1.15, "mid": 4.18, "hi": 6.61 }
079ee0
sequence_lucas_compute_v1_1439011603_348
Let $n$ be the number of integers $t$ such that $28 \leq t \leq 58$ and there exist positive integers $a$ and $b$ with $1 \leq a \leq 4$, $1 \leq b \leq 6$, and $t = 5a + 3b + 20$. Compute the $n$-th Lucas number.
64,079
graphs = [ Graph( let={ "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...
NT
null
COMPUTE
sympy
LIN_FORM
[ "LIN_FORM" ]
7b2633
sequence_lucas_compute_v1
null
5
0
[ "LIN_FORM" ]
1
0.003
2026-02-08T15:25:28.658617Z
{ "verified": true, "answer": 64079, "timestamp": "2026-02-08T15:25:28.661234Z" }
599c63
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 128, "completion_tokens": 1727 }, "timestamp": "2026-02-16T06:27:25.768Z", "answer": 64079 }, ...
1
[ { "lemma": "K13", "status": "no" }, { "lemma": "K15", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" }, { "lemma": "MOD_ADD", "status": "no" }, { "lemma": "V7", "status": "no" } ]
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
f5cf0f
nt_lcm_compute_v1_1353956133_558
Let $a$ be the largest prime number less than or equal to $2372$. Let $b = 1647$. Define $\text{result}$ to be the least common multiple of $a$ and $b$. Let $Q$ be the remainder when $\text{result}$ is divided by $88959$. Compute $Q$.
79,800
graphs = [ Graph( let={ "_n": Const(2), "a": MaxOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Ref("_n")), Leq(Var("n"), Const(2372)), IsPrime(Var("n"))))), "b": Const(1647), "result": LCM(a=Ref("a"), b=Ref("b")), "Q": Mod(value...
NT
null
COMPUTE
sympy
MAX_PRIME_BELOW
[ "MAX_PRIME_BELOW" ]
dc3ad3
nt_lcm_compute_v1
null
3
0
[ "MAX_PRIME_BELOW" ]
1
0.001
2026-02-08T11:31:24.915964Z
{ "verified": true, "answer": 79800, "timestamp": "2026-02-08T11:31:24.917171Z" }
4ae74e
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 117, "completion_tokens": 1091 }, "timestamp": "2026-02-14T17:46:33.223Z", "answer": 79800 }, ...
1
[ { "lemma": "K18", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok" }, { "lemma": "MOD_ADD", "status": "no" }, { "lemma": "V8", "status": "no" }, { "lemma": "V8_SUM", "status": "no" } ]
{ "lo": -5.14, "mid": 0.32, "hi": 6.51 }
5152a9
comb_count_permutations_fixed_v1_2051736721_5584
Let $n = 9$ and $k = \sum_{i=1}^{2} \phi(i) \cdot \left\lfloor \frac{2}{i} \right\rfloor$, where $\phi$ is Euler's totient function. Compute $\binom{n}{k} \cdot !(n - k)$, where $!m$ denotes the number of derangements of $m$ elements.
22,260
graphs = [ Graph( let={ "_n": Const(2), "n": Const(9), "k": Summation(var="k1", start=Const(1), end=Ref("_n"), expr=Mul(EulerPhi(n=Var("k1")), Floor(Div(Const(2), Var("k1"))))), "result": Mul(Binom(n=Ref("n"), k=Ref("k")), Subfactorial(arg=Sub(left=Ref(name='n...
NT
COMB
COUNT
sympy
K2
[ "K2" ]
6897ab
comb_count_permutations_fixed_v1
null
5
0
[ "K2" ]
1
0.002
2026-02-08T18:40:40.705058Z
{ "verified": true, "answer": 22260, "timestamp": "2026-02-08T18:40:40.706585Z" }
b75ff1
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 131, "completion_tokens": 1355 }, "timestamp": "2026-02-18T18:35:48.405Z", "answer": 22260 }, ...
1
[ { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "K15", "status": "no" }, { "lemma": "K2", "status": "ok" }, { "lemma": "L3c", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" }, { "lemma": "V8", "status": "no" } ]
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
b69996
antilemma_k2_v1_2051736721_3356
Let $S$ be the set of all integers $x$ such that $$x^2-80x+1431=0.$$ Let $A$ be the sum of all elements of $S$. Let $$C=\sum_{d\mid 50} \varphi(d),$$ where $\varphi$ denotes Euler's totient function. Define $$N=\frac{80}{A}.$$ Let $$x=\sum_{k=N}^{50} \varphi(k)\left\lfloor \frac{C}{k} \right\rfloor.$$ Compute $x$.
1,275
graphs = [ Graph( let={ "_m": Const(1431), "_n": Const(2), "x": Summation(var="k", start=Div(Const(80), SumOverSet(set=SolutionsSet(var=Var("x1"), condition=Eq(Sum(Pow(Var("x1"), Ref("_n")), Mul(Const(-80), Var("x1")), Ref("_m")), Sub(Const(23), Const(23)))))), end=Const(...
NT
COMB
COMPUTE
sympy
VIETA_SUM
[ "VIETA_SUM/IDENTITY_DIV_SELF/K2", "IDENTITY_SUB_SELF", "K3/K2", "K2" ]
8f0b6c
antilemma_k2_v1
null
5
0
[ "IDENTITY_DIV_SELF", "IDENTITY_SUB_SELF", "K2", "K3", "VIETA_SUM" ]
5
0.006
2026-02-08T17:16:14.247407Z
{ "verified": true, "answer": 1275, "timestamp": "2026-02-08T17:16:14.253167Z" }
104439
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 165, "completion_tokens": 925 }, "timestamp": "2026-02-17T22:54:13.489Z", "answer": 1275 }, { ...
1
[ { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "IDENTITY_DIV_SELF", "status": "ok_later" }, { "lemma": "IDENTITY_SUB_SELF", "status": "ok" }, { "lemma": "K14", "status": "no" }, { "lem...
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
f1bb2d
modular_mod_compute_v1_2051736721_3523
Let $n = 100$. Consider the set of all ordered pairs $(x, y)$ of positive integers such that $x + y = n$. Define $m$ to be the maximum value of $xy$ over all such pairs. Compute the remainder when $-64$ is divided by $m$.
2,436
graphs = [ Graph( let={ "_n": Const(100), "a": Const(-64), "m": 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")), Ref("_n")))), expr=...
NT
null
COMPUTE
sympy
B1
[ "B1" ]
5b950e
modular_mod_compute_v1
null
3
0
[ "B1" ]
1
0.002
2026-02-08T17:23:33.106108Z
{ "verified": true, "answer": 2436, "timestamp": "2026-02-08T17:23:33.107748Z" }
78af54
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 109, "completion_tokens": 553 }, "timestamp": "2026-02-18T01:03:02.028Z", "answer": 2436 }, { ...
1
[ { "lemma": "B1", "status": "ok" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", "status": "no" }, { "lemma": "VAL_SUM_EQ", "status": "no" } ]
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
49c75e
nt_min_crt_v1_151522320_2134
Let $T$ be the set of all integers $t$ such that $7 \le t \le 20$ and there exist positive integers $a \le 5$ and $b \le 2$ for which $t = 2a + 5b$. Let $u = |T|$. Define $n$ to be the smallest positive integer such that $n \le u$, $n \equiv 2 \pmod{5}$, and $n \equiv 2 \pmod{11}$. Find the remainder when $|n|$ is divi...
2
graphs = [ Graph( let={ "m": Const(5), "k": Const(11), "a": Const(2), "b": Const(2), "upper": Summation(var="k", start=Const(1), end=CountOverSet(set=SolutionsSet(var=Var("t"), condition=Exists(var=Var(name='a'), condition=Exists(var=Var(name='b'),...
NT
null
EXTREMUM
sympy
MIN_PRIME_FACTOR
[ "LIN_FORM/SUM_ARITHMETIC" ]
5a2696
nt_min_crt_v1
null
5
0
[ "LIN_FORM", "MIN_PRIME_FACTOR", "SUM_ARITHMETIC" ]
3
0.164
2026-02-08T04:37:49.987698Z
{ "verified": true, "answer": 2, "timestamp": "2026-02-08T04:37:50.151259Z" }
addd88
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 239, "completion_tokens": 813 }, "timestamp": "2026-02-11T21:38:30.647Z", "answer": 2 }, { "id":...
1
[ { "lemma": "K14", "status": "no" }, { "lemma": "K15", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "L3c", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "SUM_ARIT...
{ "lo": -7.01, "mid": -4.68, "hi": -2.15 }
76fd1c
nt_count_divisors_in_range_v1_458359167_102
Let $S$ be the set of all positive integers $t$ such that $5 \leq t \leq 35$ and there exist positive integers $a$, $b$ with $1 \leq a \leq 13$, $1 \leq b \leq 3$, and $t = 2a + 3b$. Let $m = 19$ and $n = 332640$. Let $a$ be the largest integer such that $19^a$ divides $19^{25}$. Let $b$ be the largest integer such tha...
128
graphs = [ Graph( let={ "_m": Const(19), "_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=13)), Geq(left=Var...
NT
null
COUNT
sympy
LIN_FORM
[ "LIN_FORM/K13", "K14" ]
5e8c98
nt_count_divisors_in_range_v1
null
4
0
[ "K13", "K14", "LIN_FORM" ]
3
0.128
2026-02-08T02:59:29.204108Z
{ "verified": true, "answer": 128, "timestamp": "2026-02-08T02:59:29.331688Z" }
82c428
CC BY 4.0
[ { "id": 8, "model": "mathstral", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 3829, "completion_tokens": 556 }, "timestamp": "2026-02-17T17:45:47.233Z", "answer": 128 } ]
2
[ { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "K13", "status": "ok_later" }, { "lemma": "K14", "status": "ok" }, { "lemma": "K5", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "LTE_DIFF", "status": "no" }, ...
{ "lo": -6.48, "mid": 1.54, "hi": 9.56 }
03b2b7
antilemma_cartesian_v1_784195855_2580
Compute the remainder when $27 - (33 \times 43)$ is divided by $90663$.
89,271
graphs = [ Graph( let={ "x": CountOverSet(set=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(33)), right=IntegerRange(start=Const(1), end=Const(43)))), "_c": Const(27), "Q": Mod(value=Sub(Ref("_c"), Ref("x")), modulus=Const(90663)), }, goal=R...
COMB
GEOM
COMPUTE
sympy
COUNT_CARTESIAN
[ "COUNT_CARTESIAN" ]
409d7d
antilemma_cartesian_v1
null
2
0
[ "COUNT_CARTESIAN" ]
1
0.001
2026-02-08T05:52:57.098353Z
{ "verified": true, "answer": 89271, "timestamp": "2026-02-08T05:52:57.099055Z" }
a04332
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 131, "completion_tokens": 317 }, "timestamp": "2026-02-24T04:47:08.995Z", "answer": 89271 }, { "i...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_CARTESIAN", "status": "ok" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", "status": "no" }, { "lemma": "V8_SUM", "status": "no" } ]
{ "lo": -7.18, "mid": -5, "hi": -3.01 }
4633e9
comb_binomial_compute_v1_1218484723_1864
Let $n$ be the number of ordered pairs $(a, b)$ of positive integers with $1 \le a, b \le 30$ such that $17a^4 + 68a^3b + 102a^2b^2 + 68ab^3 + 17b^4 = 860625$. Compute $\binom{n}{7}$.
3,432
graphs = [ Graph( let={ "_n": Const(4), "n": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var("a"), Var("b")]), condition=And(Geq(Var("a"), Const(1)), Leq(Var("a"), Const(30)), Geq(Var("b"), Const(1)), Leq(Var("b"), Const(30)), Eq(Sum(Mul(Const(68), Var("a"), Pow(Var("b"), Const...
COMB
null
COMPUTE
sympy
POLY4_COUNT
[ "POLY4_COUNT" ]
861d91
comb_binomial_compute_v1
null
3
0
[ "POLY4_COUNT" ]
1
0.002
2026-02-25T03:33:16.241049Z
{ "verified": true, "answer": 3432, "timestamp": "2026-02-25T03:33:16.242711Z" }
ca8eb1
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 191, "completion_tokens": 1095 }, "timestamp": "2026-03-29T01:48:48.956Z", "answer": 3432 }, { "i...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "POLY4_COUNT", "status": "ok" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8_SUM", "status": "no" ...
{ "lo": -4.26, "mid": -1.81, "hi": 1.23 }
790b7a
alg_qf_psd_count_v1_1218484723_437
Let $A = \min\left\{ 64a_1^3 + 152b_1^3 + 108a_1b_1^2 + 144a_1^2b_1 \mid a_1, b_1 \in \mathbb{Z}^+,\, 1 \leq a_1, b_1 \leq 11 \right\}$. Find the number of ordered pairs $(a, b)$ of positive integers with $1 \leq a \leq A$ and $1 \leq b \leq 468$ such that $$18b^2 - 42ab + 29a^2 = 232713.$$
11
graphs = [ Graph( let={ "_n": Const(108), "result": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var("a"), Var("b")]), condition=And(Geq(Var("a"), Const(1)), Leq(Var("a"), MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var("a1"), Var("b1")]), condition=And(Geq(Va...
ALG
null
COUNT
sympy
B3
[ "POLY3_MIN" ]
e2e279
alg_qf_psd_count_v1
null
5
0
[ "B3", "POLY3_MIN" ]
2
4.265
2026-02-25T02:08:22.480918Z
{ "verified": true, "answer": 11, "timestamp": "2026-02-25T02:08:26.746015Z" }
45b98b
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 260, "completion_tokens": 12627 }, "timestamp": "2026-03-28T22:38:07.548Z", "answer": 11 }, { "id...
1
[ { "lemma": "POLY3_MIN", "status": "ok" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" } ]
{ "lo": 1.22, "mid": 3.76, "hi": 5.73 }
8082c9
comb_factorial_compute_v1_677425708_1382
Let $n$ be the number of positive integers $p$ such that there exists a positive integer $q$ with $p < q$, $\gcd(p, q) = 1$, and $p \cdot q = 6300$. Let $f = n!$ denote the factorial of $n$. Compute the remainder when $68253 \cdot f$ is divided by 67742.
9,952
graphs = [ Graph( let={ "_n": Const(67742), "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=6300)), Eq(left=GCD(a=Var(name='p'), b=Var(na...
NT
null
COMPUTE
sympy
COPRIME_PAIRS
[ "COPRIME_PAIRS" ]
2bb3aa
comb_factorial_compute_v1
null
5
0
[ "COPRIME_PAIRS" ]
1
0.002
2026-02-08T04:08:47.862772Z
{ "verified": true, "answer": 9952, "timestamp": "2026-02-08T04:08:47.864363Z" }
f25472
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 196, "completion_tokens": 2335 }, "timestamp": "2026-02-09T19:13:46.081Z", "answer": 9952 }, { "i...
1
[ { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "K14", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" }, { "lemma": "V8_SUM", "status": "no" } ]
{ "lo": -3.53, "mid": 1.02, "hi": 5.49 }
98a33a
comb_factorial_compute_v1_1742523217_2846
Let $n$ be the number of positive integers at most 15 that are relatively prime to 14. Compute $n!$.
5,040
graphs = [ Graph( let={ "_n": Const(15), "n": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), Ref("_n")), Eq(GCD(a=Var("n"), b=Const(14)), Const(1))))), "result": Factorial(Ref("n")), }, goal=Ref("result"), ...
NT
null
COMPUTE
sympy
C4
[ "C4" ]
08d162
comb_factorial_compute_v1
null
3
0
[ "C4" ]
1
0.002
2026-02-08T05:24:46.284033Z
{ "verified": true, "answer": 5040, "timestamp": "2026-02-08T05:24:46.285704Z" }
c7b6de
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 74, "completion_tokens": 598 }, "timestamp": "2026-02-12T08:23:20.358Z", "answer": 5040 }, { ...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "C4", "status": "ok" }, { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "K13", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "MOD_SUB", "status": "no" } ]
{ "lo": -7.01, "mid": -4.68, "hi": -2.15 }
c0f865
antilemma_k2_v1_784195855_3218
Let $m = 315$. For each positive divisor $d$ of $m$, let $\phi(d)$ denote Euler's totient function evaluated at $d$. Define $n$ to be the sum of $\phi(d)$ over all positive divisors $d$ of $m$. Compute $$ \sum_{k=1}^{315} \phi(k) \left\lfloor \frac{n}{k} \right\rfloor. $$
49,770
graphs = [ Graph( let={ "_m": Const(315), "_n": SumOverDivisors(n=Const(value=315), var='d', expr=EulerPhi(n=Var(name='d'))), "x": Summation(var="k", start=Const(1), end=Ref("_m"), expr=Mul(EulerPhi(n=Var("k")), Floor(Div(Ref("_n"), Var("k"))))), }, goal=R...
NT
COMB
COMPUTE
sympy
K3
[ "K3/K2", "K2" ]
c7f244
antilemma_k2_v1
null
7
0
[ "K2", "K3" ]
2
0.001
2026-02-08T06:18:11.963284Z
{ "verified": true, "answer": 49770, "timestamp": "2026-02-08T06:18:11.964014Z" }
008fab
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 135, "completion_tokens": 971 }, "timestamp": "2026-02-12T22:22:41.957Z", "answer": 49770 }, {...
1
[ { "lemma": "K13", "status": "no" }, { "lemma": "K14", "status": "no" }, { "lemma": "K2", "status": "ok" }, { "lemma": "K3", "status": "ok" }, { "lemma": "K5", "status": "no" }, { "lemma": "MAX_VAL", "status": "no" }, { "lemma": "V3", "s...
{ "lo": -3.44, "mid": 1.34, "hi": 6.56 }
245a14
geo_visible_lattice_v1_1353956133_94
Let $n = 144$. A lattice point $(x, y)$ is visible from the origin if $\gcd(x, y) = 1$. Let $R$ be the number of visible lattice points $(x, y)$ with $1 \leq x, y \leq n$. Compute the remainder when $64833 \cdot R$ is divided by $96991$.
80,096
graphs = [ Graph( let={ "n": Const(144), "result": VisibleLatticePoints(n=Ref(name='n')), "Q": Mod(value=Mul(Const(64833), Ref("result")), modulus=Const(96991)), }, goal=Ref("Q"), ) ]
GEOM
null
COUNT
sympy
[]
geo_visible_lattice_v1
null
4
0
null
null
1.783
2026-02-08T11:18:21.189063Z
{ "verified": true, "answer": 80096, "timestamp": "2026-02-08T11:18:22.971672Z" }
fab9d9
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 191, "completion_tokens": 32768 }, "timestamp": "2026-02-24T13:21:07.007Z", "answer": null }, { ...
1
[]
{ "lo": 1.15, "mid": 4.18, "hi": 6.61 }
8acd58
comb_binomial_compute_v1_124444284_2121
Let $n = \sum_{k=1}^{5} \phi(k) \left\lfloor \frac{5}{k} \right\rfloor$, where $\phi$ denotes Euler's totient function. Let $k = 8$. Let $R = \binom{n}{k}$, and let $Q$ be the remainder when $44121 \cdot R$ is divided by 89783. Find the value of $Q$.
24,789
graphs = [ Graph( let={ "_n": Const(5), "n": Summation(var="k", start=Const(1), end=Const(5), expr=Mul(EulerPhi(n=Var("k")), Floor(Div(Ref("_n"), Var("k"))))), "k": Const(8), "result": Binom(n=Ref("n"), k=Ref("k")), "Q": Mod(value=Mul(Const(44121),...
NT
null
COMPUTE
sympy
K2
[ "K2" ]
6897ab
comb_binomial_compute_v1
null
5
0
[ "K2" ]
1
0.003
2026-02-08T04:19:38.399108Z
{ "verified": true, "answer": 24789, "timestamp": "2026-02-08T04:19:38.401683Z" }
fb7cd4
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 207, "completion_tokens": 1248 }, "timestamp": "2026-02-10T16:32:04.979Z", "answer": 24789 }, { ...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "K2", "status": "ok" }, { "lemma": "L3c", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "V8", "status": "no" } ]
{ "lo": -3.42, "mid": 1.86, "hi": 7.55 }
f55143
modular_modexp_compute_v1_1742523217_1313
Let $ m = 12 $. Define $ S $ to be the set of all positive integers $ p $ for which there exists a positive integer $ q $ such that $ p \cdot q = 216 $, $ \gcd(p, q) = 1 $, and $ p < q $. Let $ n = |S| $. Let $ a $ be the largest prime number $ r $ such that $ n \leq r \leq 12 $. Define $ T $ to be the set of all order...
9,483
graphs = [ Graph( let={ "_m": Const(12), "_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=216)), Eq(left=GCD(a=Var(name='p'), b=Var(name=...
NT
null
COMPUTE
sympy
COPRIME_PAIRS
[ "COPRIME_PAIRS/MAX_PRIME_BELOW", "B3" ]
fdc414
modular_modexp_compute_v1
null
6
0
[ "B3", "COPRIME_PAIRS", "MAX_PRIME_BELOW" ]
3
0.003
2026-02-08T03:40:43.292070Z
{ "verified": true, "answer": 9483, "timestamp": "2026-02-08T03:40:43.295000Z" }
41f6b0
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 281, "completion_tokens": 9170 }, "timestamp": "2026-02-23T21:10:18.415Z", "answer": 9483 }, { "...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "C2", "status": "no" }, { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "K13", "status": "no" }, { "lemma": "K17", "status": "no" }, { ...
{ "lo": 1.96, "mid": 5.23, "hi": 8.52 }
2fb169
comb_bell_compute_v1_1742523217_529
Let $m = 8456$. Let $T$ be the set of all ordered pairs $(x, y)$ of positive integers such that $xy = 17875984$. Let $s$ be the minimum value of $x + y$ over all such pairs. Let $j$ range over the nonnegative integers from 0 to $s$, inclusive. Let $n$ be the number of such $j$ for which $\binom{m}{j}$ is odd. Let $B_n$...
54
graphs = [ Graph( let={ "_m": Const(8456), "_n": Const(2), "n": CountOverSet(set=SolutionsSet(var=Var("j"), condition=And(Geq(Var("j"), Const(0)), Leq(Var("j"), MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var("x"), Var("y")]), condition=And(IsPositive(a...
NT
COMB
COMPUTE
sympy
LIN_FORM
[ "B3/V8" ]
4fad5b
comb_bell_compute_v1
null
7
0
[ "B3", "LIN_FORM", "V8" ]
3
0.024
2026-02-08T03:06:47.476022Z
{ "verified": true, "answer": 54, "timestamp": "2026-02-08T03:06:47.500016Z" }
d93350
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 270, "completion_tokens": 3567 }, "timestamp": "2026-02-09T04:05:37.364Z", "answer": 54 }, { "id...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "K14", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MOD_SUB", "status": "no" }, ...
{ "lo": 1.96, "mid": 5.23, "hi": 8.52 }
967921
alg_sym_quad_system_v1_601307018_5798
Let $M$ be the sum of $a^5 + b^5 + c^5$ over all positive integers $a, b, c$ satisfying $a^2 + b^2 + c^2 = ab + bc + ca$ and $2a + 4b + 3c = 2349$, taken modulo the number of pairs $(a_1, b_1)$ with $1 \le a_1, b_1 \le 40$ such that $41a_1^2 - 12a_1b_1 + 20b_1^2 \le 34697$. Find the remainder when $44121M$ is divided b...
31,732
graphs = [ Graph( let={ "_n": Const(2), "result": Mod(value=SumOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var("a"), Var("b"), Var("c")]), condition=And(Eq(Sum(Pow(Var("a"), Const(2)), Pow(Var("b"), Const(2)), Pow(Var("c"), Const(2))), Sum(Mul(Var("a"), Var("b")), Mul...
ALG
null
COMPUTE
sympy
QF_PSD_COUNT_LEQ
[ "QF_PSD_COUNT_LEQ" ]
009d0f
alg_sym_quad_system_v1
null
7
0
[ "QF_PSD_COUNT_LEQ" ]
1
0.023
2026-03-10T06:20:21.755401Z
{ "verified": true, "answer": 31732, "timestamp": "2026-03-10T06:20:21.778658Z" }
442af8
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 259, "completion_tokens": 11317 }, "timestamp": "2026-04-19T02:57:03.494Z", "answer": 31732 }, { ...
1
[ { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" }, { "lemma": "QF_PSD_COUNT_LEQ", "status": "ok" } ]
{ "lo": 2.1, "mid": 5.36, "hi": 8.63 }
fc7b6c
modular_sum_quadratic_residues_v1_458359167_1549
Let $p = 401$. Let $S$ be the set of all positive integers $p$ such that there exists a positive integer $q$ satisfying $p \cdot q = 150$, $\gcd(p, q) = 1$, and $p < q$. Compute $\frac{p(p-1)}{|S|}$.
40,100
graphs = [ Graph( let={ "p": Const(401), "result": Div(Mul(Ref("p"), Sub(Ref("p"), Const(1))), 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...
NT
null
SUM
sympy
COPRIME_PAIRS
[ "COPRIME_PAIRS" ]
2bb3aa
modular_sum_quadratic_residues_v1
null
4
0
[ "COPRIME_PAIRS" ]
1
0.001
2026-02-08T04:45:16.581484Z
{ "verified": true, "answer": 40100, "timestamp": "2026-02-08T04:45:16.582778Z" }
77a237
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 183, "completion_tokens": 1389 }, "timestamp": "2026-02-11T21:52:01.129Z", "answer": 40100 }, { ...
1
[ { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "L3c", "status": "no" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" }, { "lemma": "V1", "status": ...
{ "lo": -3.42, "mid": 1.86, "hi": 7.55 }
80e0bc
comb_count_partitions_v1_865884756_4030
Let $T$ be the set of all integers $t$ such that $20 \leq t \leq 122$ and $t = 14a + 6b$ for some integers $a$ and $b$ with $1 \leq a \leq 4$ and $1 \leq b \leq 11$. Let $n$ be the number of elements in $T$. Compute the number of integer partitions of $n$.
37,338
graphs = [ Graph( let={ "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...
COMB
null
COUNT
sympy
LIN_FORM
[ "LIN_FORM" ]
7b2633
comb_count_partitions_v1
null
5
0
[ "LIN_FORM" ]
1
0.003
2026-02-08T17:43:02.888635Z
{ "verified": true, "answer": 37338, "timestamp": "2026-02-08T17:43:02.891331Z" }
52b4cb
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 139, "completion_tokens": 1861 }, "timestamp": "2026-02-18T06:41:47.870Z", "answer": 37338 }, ...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", "status": "no" }, { "lemma": "V8_SUM", "status": "no" } ]
{ "lo": -2.38, "mid": 1.74, "hi": 6.59 }
f66d60
modular_count_residue_v1_458359167_1348
Let $m = 7$. Define $n$ to be the number of prime numbers $p$ such that $2 \leq p \leq m$. Let $r$ be the maximum value of $xy$ over all pairs of positive integers $(x, y)$ such that $x + y = n$. Let $t$ range over integers from 18 to 51, and define $m'$ to be the number of values of $t$ for which there exist positive ...
93,680
graphs = [ Graph( let={ "_m": Const(7), "_n": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(2)), Leq(Var("n"), Ref("_m")), IsPrime(Var("n"))))), "upper": Const(34596), "m": CountOverSet(set=SolutionsSet(var=Var("t"), condition=E...
NT
null
COUNT
sympy
COUNT_PRIMES
[ "COUNT_PRIMES/LIN_FORM/B1" ]
e80d65
modular_count_residue_v1
null
5
0
[ "B1", "COUNT_PRIMES", "LIN_FORM" ]
3
1.426
2026-02-08T04:33:34.228946Z
{ "verified": true, "answer": 93680, "timestamp": "2026-02-08T04:33:35.654513Z" }
3e994e
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 299, "completion_tokens": 1713 }, "timestamp": "2026-02-10T17:01:58.439Z", "answer": 93680 }, { ...
1
[ { "lemma": "B1", "status": "ok_later" }, { "lemma": "COUNT_PRIMES", "status": "ok" }, { "lemma": "K5", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok_later" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "V5", "status": "no" },...
{ "lo": -3.49, "mid": 1.84, "hi": 7.55 }
9427b4
comb_binomial_compute_v1_1520064083_2928
Let $n = 12$. Let $k$ be the sum of all real solutions $x$ to the equation $x^2 - 7x - 1548 = 0$. Compute $\binom{n}{k}$.
792
graphs = [ Graph( let={ "n": Const(12), "k": SumOverSet(set=SolutionsSet(var=Var("x"), condition=Eq(Sum(Pow(Var("x"), Const(2)), Mul(Const(-7), Var("x")), Const(-1548)), Const(0)))), "result": Binom(n=Ref("n"), k=Ref("k")), }, goal=Ref("result"), ) ]
ALG
COMB
COMPUTE
sympy
VIETA_SUM
[ "VIETA_SUM" ]
b33a7a
comb_binomial_compute_v1
null
3
0
[ "VIETA_SUM" ]
1
0.002
2026-02-08T05:19:13.840760Z
{ "verified": true, "answer": 792, "timestamp": "2026-02-08T05:19:13.842623Z" }
f8a478
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 160, "completion_tokens": 574 }, "timestamp": "2026-02-24T03:16:18.216Z", "answer": 792 }, { "id"...
1
[ { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" }, { "lemma": "VIETA_SUM", "status": "ok" } ]
{ "lo": -9.23, "mid": -6.18, "hi": -4.06 }
9f3d81
comb_catalan_compute_v1_168721529_374
Let $T$ be the set of all positive integers $t$ such that $9 \le t \le 36$ and there exist positive integers $a$ and $b$ with $1 \le a \le 2$, $1 \le b \le 11$, and $t = 7a + 2b$. Let $N$ be the number of elements in $T$. Now consider the set of all ordered pairs $(x_1, x_2)$ of positive odd integers such that $x_1 + x...
58,786
graphs = [ Graph( let={ "_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=2)), Geq(left=Var(name='b'), right=Const(value=...
COMB
null
COMPUTE
sympy
LIN_FORM
[ "LIN_FORM/COMB1" ]
268a62
comb_catalan_compute_v1
null
7
0
[ "COMB1", "LIN_FORM" ]
2
0.003
2026-02-08T13:00:53.618747Z
{ "verified": true, "answer": 58786, "timestamp": "2026-02-08T13:00:53.621290Z" }
34f7c1
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 251, "completion_tokens": 1734 }, "timestamp": "2026-02-24T16:58:06.390Z", "answer": 58786 }, { "...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COMB1", "status": "ok_later" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "V8", "status": "...
{ "lo": -2.77, "mid": -0.49, "hi": 2.44 }
6b00ec
modular_inverse_v1_458359167_2017
Let $a = \sum_{k=1}^{12} k$. Let $x$ be the smallest positive integer such that $1 \leq x \leq 112$ and $a \cdot x \equiv 1 \pmod{113}$. Compute the remainder when $5^{|x|} + 16129$ is divided by $99991$.
45,395
graphs = [ Graph( let={ "a": Summation(var="k", start=Const(1), end=Const(12), expr=Var("k")), "m": Const(113), "upper": Const(112), "result": MinOverSet(set=SolutionsSet(var=Var("x"), condition=And(Geq(Var("x"), Const(1)), Leq(Var("x"), Ref("upper")), Eq(Mod(...
ALG
NT
EXTREMUM
sympy
SUM_ARITHMETIC
[ "SUM_ARITHMETIC" ]
eb34f0
modular_inverse_v1
null
6
0
[ "SUM_ARITHMETIC" ]
1
0.016
2026-02-08T04:58:50.449045Z
{ "verified": true, "answer": 45395, "timestamp": "2026-02-08T04:58:50.465154Z" }
d41c23
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 190, "completion_tokens": 6887 }, "timestamp": "2026-02-11T22:33:16.740Z", "answer": 45395 }, { ...
1
[ { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" }, { "lemma": "SUM_ARITHMETIC", "status": "ok" } ]
{ "lo": -3.42, "mid": 1.86, "hi": 7.55 }
65e8be
comb_count_permutations_fixed_v1_1520064083_990
Let $n = 10$ and $k = 5$. Define $\text{result} = \binom{n}{k} \cdot !(n - k)$, where $!m$ denotes the number of derangements of $m$ objects. Let $c = 26393$ and let $Q$ be the remainder when $c \cdot \text{result}$ is divided by $93690$. Compute $Q$.
51,714
graphs = [ Graph( let={ "n": Const(10), "k": Const(5), "result": Mul(Binom(n=Ref("n"), k=Ref("k")), Subfactorial(arg=Sub(left=Ref(name='n'), right=Ref(name='k')))), "_c": Const(26393), "Q": Mod(value=Mul(Ref("_c"), Ref("result")), modulus=Const(936...
COMB
null
COUNT
sympy
LIN_FORM
[ "LIN_FORM/MOBIUS_COPRIME" ]
d91cf7
comb_count_permutations_fixed_v1
null
4
0
[ "LIN_FORM", "MOBIUS_COPRIME" ]
2
0.073
2026-02-08T03:42:01.323261Z
{ "verified": true, "answer": 51714, "timestamp": "2026-02-08T03:42:01.395830Z" }
d54754
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 203, "completion_tokens": 1440 }, "timestamp": "2026-02-10T15:35:43.723Z", "answer": 51714 }, { "...
1
[ { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": ...
{ "lo": -2.46, "mid": 0.47, "hi": 3.59 }
75e581
comb_sum_binomial_mod_v1_1080341949_337
Let $S$ be the set of all ordered pairs $(x, y)$ of positive integers such that $xy = 961$. Let $m$ be the minimum value of $x + y$ over all pairs in $S$. Compute the remainder when $\sum_{k=1}^{41} \binom{m}{k}$ is divided by $10567$.
739
graphs = [ Graph( let={ "_n": Const(41), "sum": Summation(var="k", start=Const(1), end=Ref("_n"), expr=Binom(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("...
ALG
COMB
COMPUTE
sympy
B3
[ "B3" ]
0cd20d
comb_sum_binomial_mod_v1
null
5
0
[ "B3" ]
1
0.01
2026-02-08T13:25:48.429963Z
{ "verified": true, "answer": 739, "timestamp": "2026-02-08T13:25:48.439726Z" }
368748
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 1, "correct": { "strict": false, "boxed": false, "relaxed": true }, "usage": { "prompt_tokens": 186, "completion_tokens": 32768 }, "timestamp": "2026-02-24T18:28:29.431Z", "answer": null }, { ...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" } ]
{ "lo": 3.25, "mid": 5.68, "hi": 8.81 }
d7cb65
comb_count_surjections_v1_458359167_3854
Let $n$ be the number of ordered pairs $(i, j)$ of positive integers such that $1 \leq i \leq 6$, $1 \leq j \leq 7$, and $i + j = 7$. Let $k = 4$. Define $\text{result} = k! \cdot S(n, k)$, where $S(n, k)$ denotes the Stirling number of the second kind. Compute $\text{result}$.
1,560
graphs = [ Graph( let={ "n": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var("i"), Var("j")]), condition=Eq(Sum(Var("i"), Var("j")), Const(7)), domain=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(6)), right=IntegerRange(start=Const(1), end=Const(7))))), "k": Con...
COMB
null
COUNT
sympy
COUNT_SUM_EQUALS
[ "COUNT_SUM_EQUALS" ]
75ab0f
comb_count_surjections_v1
null
4
0
[ "COUNT_SUM_EQUALS" ]
1
0.016
2026-02-08T11:23:52.975154Z
{ "verified": true, "answer": 1560, "timestamp": "2026-02-08T11:23:52.990899Z" }
fc2f75
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 212, "completion_tokens": 862 }, "timestamp": "2026-02-24T13:44:57.701Z", "answer": 1560 }, { "id...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "ok" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", "status": "no" }, { "lemma": "V8_SUM", "status": "no" } ]
{ "lo": -3.84, "mid": -1.67, "hi": 1.32 }
664d79
modular_mod_compute_v1_677425708_3664
Let $T = \sum_{k=1}^{10} k$, and let $a = \sum_{k=1}^{T} k$. Let $m = 4489$. Compute the remainder when $a$ is divided by $m$, then compute the remainder when $50457$ times that result is divided by $94421$.
89,718
graphs = [ Graph( let={ "_n": Summation(var="k", start=Const(1), end=Const(10), expr=Var("k")), "a": Summation(var="k", start=Const(1), end=Ref("_n"), expr=Var("k")), "m": Const(4489), "result": Mod(value=Ref("a"), modulus=Ref("m")), "_c": Const(50...
NT
null
COMPUTE
sympy
SUM_ARITHMETIC
[ "SUM_ARITHMETIC/SUM_ARITHMETIC" ]
2a57af
modular_mod_compute_v1
null
2
0
[ "SUM_ARITHMETIC" ]
1
0.001
2026-02-08T05:53:10.386517Z
{ "verified": true, "answer": 89718, "timestamp": "2026-02-08T05:53:10.387776Z" }
05db21
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 118, "completion_tokens": 2030 }, "timestamp": "2026-02-12T15:53:50.606Z", "answer": 89718 }, ...
1
[ { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "K14", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "MOD_SUB", "status": "no" }, { "lemma": "SUM_ARITHMETIC", "status": "ok" ...
{ "lo": -3.44, "mid": 1.34, "hi": 6.56 }
fffeca
modular_sum_quadratic_residues_v1_153355830_256
Let $C = 4$. Let $m$ be the minimum value of $x + y$ over all ordered pairs $(x, y)$ of positive integers such that $xy = C$. Let $n$ be the maximum value of $xy$ over all ordered pairs $(x, y)$ of positive integers such that $x + y = m$. Let $p$ be the number of integers $t$ with $9 \leq t \leq 369$ for which there ex...
30,363
graphs = [ Graph( let={ "_c": Const(4), "_m": 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("_c")))), expr=Sum(Var("x"), Var("y")))), ...
NT
null
SUM
sympy
C3
[ "B3/B1/LIN_FORM" ]
f992df
modular_sum_quadratic_residues_v1
null
6
0
[ "B1", "B3", "C3", "LIN_FORM" ]
4
0.005
2026-02-08T02:59:23.447959Z
{ "verified": true, "answer": 30363, "timestamp": "2026-02-08T02:59:23.453271Z" }
22280d
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 257, "completion_tokens": 9217 }, "timestamp": "2026-02-23T15:25:04.381Z", "answer": 30363 }, { ...
1
[ { "lemma": "B1", "status": "ok_later" }, { "lemma": "B3", "status": "ok" }, { "lemma": "C2", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok_later" }, { "lemma": "MOD_MUL", "status": "no" }, { "lem...
{ "lo": 1.96, "mid": 5.23, "hi": 8.52 }
7dc0b8_l
geo_visible_lattice_v1_1125832087_2450
Let $n = 120$. Define $V$ to be the number of lattice points $(x, y)$ with $1 \leq x, y \leq n$ such that $\gcd(x, y) = 1$. Compute the remainder when $2304 - V$ is divided by $66189$.
0
GEOM
null
COUNT
sympy
[]
geo_visible_lattice_v1
null
5
0
null
null
0.305
2026-02-08T04:37:23.450336Z
{ "verified": false, "answer": 59722, "timestamp": "2026-02-08T04:37:23.755469Z" }
2d42e5
7dc0b8
legacy_text
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 177, "completion_tokens": 9230 }, "timestamp": "2026-02-24T01:15:24.825Z", "answer": 59722 }, { "...
1
[]
{ "lo": 1.97, "mid": 4.36, "hi": 6.69 }
450860
diophantine_fbi2_min_v1_784195855_10208
Let $n = 317$, $k = 33$, and $u = 43$. Define $D$ as the set of all integers $d$ such that $3 \leq d \leq u$, $d$ divides $k$, and $\frac{k}{d} \geq 4$. Let $r$ be the smallest element of $D$. Let $p$ be the smallest prime divisor of $9720582149$. Compute $$r \bmod 317 + 7001 \cdot (r \bmod p).$$
21,006
graphs = [ Graph( let={ "_n": Const(317), "k": Const(33), "upper": Const(43), "result": MinOverSet(set=SolutionsSet(var=Var("d"), condition=And(Geq(Var("d"), Const(3)), Leq(Var("d"), Ref("upper")), Divides(divisor=Var("d"), dividend=Ref("k")), Geq(Div(Ref("k")...
NT
null
EXTREMUM
sympy
MIN_PRIME_FACTOR
[ "MIN_PRIME_FACTOR" ]
cffc20
diophantine_fbi2_min_v1
two_moduli
4
0
[ "MIN_PRIME_FACTOR" ]
1
0.005
2026-02-08T17:29:32.002546Z
{ "verified": true, "answer": 21006, "timestamp": "2026-02-08T17:29:32.007379Z" }
225b79
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 160, "completion_tokens": 2440 }, "timestamp": "2026-02-18T03:17:18.891Z", "answer": 21006 }, ...
1
[ { "lemma": "K17", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "MIN_PRIME_FACTOR", "status": "ok" }, { "lemma": "MOD_ADD", "status": "no" }, { "lemma": "MOD_MUL", "status": "no" }, { "lemma": "V7", "status": "no" } ]
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
c25e88
sequence_lucas_compute_v1_784195855_7424
Let $S$ be the set of all positive integers $p$ such that there exists a positive integer $q$ with $pq = 6$, $\gcd(p, q) = 1$, and $p < q$. Let $\text{count}$ be the number of elements in $S$. Let $n$ be the largest prime number $n$ such that $\text{count} \leq n \leq 26$. Compute the $n$-th Lucas number.
64,079
graphs = [ Graph( let={ "_n": Const(26), "n": 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'...
NT
null
COMPUTE
sympy
COPRIME_PAIRS
[ "COPRIME_PAIRS/MAX_PRIME_BELOW" ]
2248fc
sequence_lucas_compute_v1
null
3
0
[ "COPRIME_PAIRS", "MAX_PRIME_BELOW" ]
2
0.003
2026-02-08T09:16:59.357582Z
{ "verified": true, "answer": 64079, "timestamp": "2026-02-08T09:16:59.360581Z" }
db4ba2
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 145, "completion_tokens": 1079 }, "timestamp": "2026-02-14T02:32:08.151Z", "answer": 64079 }, ...
1
[ { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "L3b", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok_later" }, { "lemma": "V...
{ "lo": -5.14, "mid": 0.32, "hi": 6.51 }
7d9e2f
alg_poly4_sum_v1_1218484723_1214
Let $S$ be the set of integers $t$ expressible as $t = 9a + 6b$ for some integers $a, b$ with $1 \le a \le 1024$, $1 \le b \le 92$, and $15 \le t \le 9768$. Let $m = |S|$. Let $n$ be the number of integers $v$ with $13 \le v \le m$ for which there exist integers $a, b$ with $1 \le a, b \le 14$ such that $-6ab + 2a^2 + ...
44,228
graphs = [ Graph( let={ "_m": Const(88478), "_n": Const(17), "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(366)), Geq(Var("b"), Const(1)), Leq(Var("b"), C...
ALG
null
COMPUTE
sympy
LIN_FORM
[ "LIN_FORM/QF_PSD_DISTINCT" ]
e187db
alg_poly4_sum_v1
null
7
0
[ "LIN_FORM", "QF_PSD_DISTINCT" ]
2
0.62
2026-02-25T02:59:36.607562Z
{ "verified": true, "answer": 44228, "timestamp": "2026-02-25T02:59:37.227501Z" }
3e6018
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 328, "completion_tokens": 32768 }, "timestamp": "2026-03-29T00:15:07.516Z", "answer": null }, { ...
1
[ { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" }, { "lemma": "QF_PSD_DISTINCT", "status": "ok_later" } ]
{ "lo": 4.76, "mid": 6.79, "hi": 9.83 }
96c1c1
nt_max_prime_below_v1_1520064083_1692
Let $P$ be the number of positive integers $p$ for which there exists an integer $q$ such that $p \cdot q = 24$, $\gcd(p, q) = 1$, and $p < q$. Let $S$ be the set of prime numbers $n$ such that $$ \sum_{d \mid P} \phi(d) \leq n \leq 12544, $$ defined over positive divisors $d$ of $P$, where $\phi$ denotes Euler's toti...
12,541
graphs = [ Graph( let={ "_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=24)), Eq(left=GCD(a=Var(name='p'), b=Var(name='q')), right=Const(value=1)), ...
NT
null
EXTREMUM
sympy
COPRIME_PAIRS
[ "COPRIME_PAIRS/K3" ]
f9481c
nt_max_prime_below_v1
null
6
0
[ "COPRIME_PAIRS", "K3" ]
2
0.617
2026-02-08T04:13:06.027049Z
{ "verified": true, "answer": 12541, "timestamp": "2026-02-08T04:13:06.643934Z" }
831c6f
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 236, "completion_tokens": 1650 }, "timestamp": "2026-02-10T15:49:21.490Z", "answer": 12541 }, { ...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "K15", "status": "no" }, { "lemma": "K3", "status": "ok_later" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, ...
{ "lo": -3.42, "mid": 1.86, "hi": 7.55 }
712a2e
nt_lcm_compute_v1_458359167_1046
Let $a$ be the largest prime number between $2$ and $2442$, inclusive. Let $b = 2271$, and let $\text{result} = \text{LCM}(a, b)$. Let $Q$ be the sum of $d_i \cdot (i+1)^2$ over all digits $d_i$ of $\text{result}$ (taken in base $10$, from right to left, starting with index $i=0$), plus $22201$. Compute $Q$.
22,824
graphs = [ Graph( let={ "_n": Const(2), "a": MaxOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Ref("_n")), Leq(Var("n"), Const(2442)), IsPrime(Var("n"))))), "b": Const(2271), "result": LCM(a=Ref("a"), b=Ref("b")), "Q": Sum(Summa...
NT
null
COMPUTE
sympy
MAX_PRIME_BELOW
[ "MAX_PRIME_BELOW" ]
dc3ad3
nt_lcm_compute_v1
null
5
0
[ "MAX_PRIME_BELOW" ]
1
0.005
2026-02-08T04:14:47.119443Z
{ "verified": true, "answer": 22824, "timestamp": "2026-02-08T04:14:47.124353Z" }
f95565
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 221, "completion_tokens": 2401 }, "timestamp": "2026-02-10T15:54:08.297Z", "answer": 22824 }, { ...
1
[ { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "K13", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok" } ]
{ "lo": -3.42, "mid": 1.86, "hi": 7.55 }
517747
nt_count_divisible_v1_1520064083_5058
Let $p$ and $q$ be positive integers such that $pq = 10187100$, $\gcd(p, q) = 1$, and $p < q$. Let $d$ be the number of such integers $p$. Let $S$ be the set of all positive integers $n$ such that $n \leq 51984$ and $n$ is divisible by $d$. Let $r = |S|$. Compute $r + \phi(r+1) + \tau(r+1)$, where $\phi$ denotes Euler'...
4,465
graphs = [ Graph( let={ "upper": Const(51984), "divisor": 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=10187100)), Eq(left=GCD(a=Var(name='...
NT
null
COUNT
sympy
COPRIME_PAIRS
[ "COPRIME_PAIRS" ]
2bb3aa
nt_count_divisible_v1
null
5
0
[ "COPRIME_PAIRS" ]
1
2.918
2026-02-08T06:35:24.842396Z
{ "verified": true, "answer": 4465, "timestamp": "2026-02-08T06:35:27.760352Z" }
decf3f
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 177, "completion_tokens": 1498 }, "timestamp": "2026-02-13T02:09:07.260Z", "answer": 4465 }, {...
1
[ { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "K13", "status": "no" }, { "lemma": "L3c", "status": "no" }, { "lemma": "V5", "status": "no" ...
{ "lo": -3.44, "mid": 1.34, "hi": 6.56 }
3e9fa6
geo_visible_lattice_v1_1520064083_7969
Let $n = 89$. Define $P$ to be the number of visible lattice points $(x, y)$ such that $1 \le x, y \le n$. Compute the value of $43681 - P$.
38,770
graphs = [ Graph( let={ "n": Const(89), "result": VisibleLatticePoints(n=Ref(name='n')), "Q": Sub(Const(43681), Ref("result")), }, goal=Ref("Q"), ) ]
GEOM
null
COUNT
sympy
[]
geo_visible_lattice_v1
null
5
0
null
null
0.533
2026-02-08T09:23:22.301824Z
{ "verified": true, "answer": 38770, "timestamp": "2026-02-08T09:23:22.835163Z" }
29437a
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 157, "completion_tokens": 4618 }, "timestamp": "2026-02-24T11:18:19.377Z", "answer": 38770 }, { "...
1
[]
{ "lo": -2.35, "mid": 1.22, "hi": 4.84 }
5027d4
comb_sum_binomial_mod_v1_458359167_3132
Let $n_1 = 0$, and define $$u = \sum_{k=0}^{n_1} (-1)^k \binom{n_1}{k}.$$ Let $a = 2u$ and $b = 2$, and define $n = a + b$. Let $$h = \sum_{k=0}^{n} (-1)^k \binom{n}{k}.$$ Compute the remainder when $$\sum_{k=28}^{175} \binom{209}{k}$$ is divided by $10369 + h$. Let this remainder be $r$. Find the remainder when $32107...
6,854
graphs = [ Graph( let={ "n1": Const(0), "u": Summation(var="k", start=Const(0), end=Ref("n1"), expr=Mul(Pow(Const(-1), Var("k")), Binom(n=Ref("n1"), k=Var("k")))), "a": Mul(Const(2), Ref("u")), "b": Const(2), "n": Sum(Ref("a"), Ref("b")), ...
COMB
null
COMPUTE
sympy
BINOMIAL_ALTERNATING
[ "BINOMIAL_ALTERNATING" ]
c21569
comb_sum_binomial_mod_v1
null
5
2
[ "BINOMIAL_ALTERNATING" ]
1
0.016
2026-02-08T06:59:19.174947Z
{ "verified": true, "answer": 6854, "timestamp": "2026-02-08T06:59:19.191380Z" }
2ff205
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 258, "completion_tokens": 32768 }, "timestamp": "2026-02-24T07:31:28.490Z", "answer": null }, { ...
1
[ { "lemma": "BINOMIAL_ALTERNATING", "status": "ok" }, { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", "status": "no" }, { "lemma": "V8_SUM", "stat...
{ "lo": 3.25, "mid": 5.68, "hi": 8.81 }
d75930
sequence_fibonacci_compute_v1_1978505735_7528
Let $n$ be the minimum value of $x + y$ over all pairs of positive integers $(x, y)$ such that $xy = 100$. Let $F_n$ denote the $n$th Fibonacci number, with $F_1 = F_2 = 1$ and $F_{k} = F_{k-1} + F_{k-2}$ for $k \geq 3$. Compute the remainder when $44121 \cdot F_n$ is divided by $89458$.
46,677
graphs = [ Graph( let={ "_n": Const(100), "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")), Ref("_n")))), expr=Sum(Var("x"), Var("y")))), ...
NT
null
COMPUTE
sympy
B3
[ "B3" ]
0cd20d
sequence_fibonacci_compute_v1
null
3
0
[ "B3" ]
1
0.002
2026-02-08T20:18:15.836143Z
{ "verified": true, "answer": 46677, "timestamp": "2026-02-08T20:18:15.837818Z" }
3c13d1
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 154, "completion_tokens": 1974 }, "timestamp": "2026-02-19T00:20:16.888Z", "answer": 46677 }, ...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, { "lemma": "MOD_MUL", "status": "no" }, { "lemma": "V3", "status": "no" }, { "lemma": "V7", "status": "no" } ]
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
240035
nt_euler_phi_compute_v1_677425708_820
Let $A$ be the set of all ordered pairs $(p,q)$ of positive integers such that $pq=36$, $\gcd(p,q)=1$, and $p<q$. Let $r$ be the number of positive integers $p$ for which there exists a positive integer $q$ with $(p,q)\in A$. Let $B$ be the set of all integers $n$ such that $n\ge r$, $n\le 4$, and $n$ is prime, and as...
73,170
graphs = [ Graph( let={ "_m": Const(73441), "_n": Const(4), "p": 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=...
NT
null
COMPUTE
sympy
COPRIME_PAIRS
[ "COPRIME_PAIRS/MAX_PRIME_BELOW/BIG_OMEGA_ONE", "MOBIUS_SUM" ]
f3a19b
nt_euler_phi_compute_v1
null
6
2
[ "BIG_OMEGA_ONE", "COPRIME_PAIRS", "MAX_PRIME_BELOW", "MOBIUS_SUM" ]
4
0.004
2026-02-08T03:46:53.086994Z
{ "verified": true, "answer": 73170, "timestamp": "2026-02-08T03:46:53.090541Z" }
ec3935
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 347, "completion_tokens": 2625 }, "timestamp": "2026-02-09T12:50:20.638Z", "answer": 73170 }, { "...
1
[ { "lemma": "BIG_OMEGA_ONE", "status": "ok_later" }, { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "K13", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MAX_PRIME_BELO...
{ "lo": -6.5, "mid": -0.2, "hi": 6.11 }
25d938
nt_count_divisible_and_v1_898971024_757
Let $S$ be the set of all integers $n$ such that $1 \leq n \leq 29976$, $n$ is divisible by $6$, and the remainder when $n$ is divided by $8$ equals $\sum_{k=0}^{1} (-1)^k \binom{1}{k}$. Find the number of elements in $S$.
1,249
graphs = [ Graph( let={ "upper": Const(29976), "d1": Const(6), "d2": Const(8), "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(Mo...
COMB
null
COUNT
sympy
BINOMIAL_ALTERNATING
[ "BINOMIAL_ALTERNATING" ]
c21569
nt_count_divisible_and_v1
null
3
0
[ "BINOMIAL_ALTERNATING" ]
1
2.152
2026-02-08T15:37:54.381120Z
{ "verified": true, "answer": 1249, "timestamp": "2026-02-08T15:37:56.533510Z" }
a59c49
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 192, "completion_tokens": 607 }, "timestamp": "2026-02-24T18:25:22.495Z", "answer": 1249 }, { "i...
1
[ { "lemma": "BINOMIAL_ALTERNATING", "status": "ok" }, { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", "status"...
{ "lo": -8.52, "mid": -5.43, "hi": -3.2 }
2a351e
modular_mod_compute_v1_784195855_2734
Let $T$ be the set of all integers $t$ such that $10 \leq t \leq 1035$ and there exist positive integers $a$ and $b$ with $1 \leq a \leq 22$, $1 \leq b \leq 482$, and $$t = 3a + 2b + 5.$$ Compute the remainder when $59536$ is divided by the number of elements in $T$.
144
graphs = [ Graph( let={ "a": Const(59536), "m": 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=22)), Geq(left=Va...
NT
null
COMPUTE
sympy
LIN_FORM
[ "LIN_FORM" ]
7b2633
modular_mod_compute_v1
null
6
0
[ "LIN_FORM" ]
1
0.003
2026-02-08T05:57:12.231347Z
{ "verified": true, "answer": 144, "timestamp": "2026-02-08T05:57:12.234082Z" }
c3aea1
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 141, "completion_tokens": 4983 }, "timestamp": "2026-02-12T17:12:06.383Z", "answer": 144 }, { ...
1
[ { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "K18", "status": "no" }, { "lemma": "L3b", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "no...
{ "lo": -3.44, "mid": 1.34, "hi": 6.56 }
548e84_n
comb_count_derangements_v1_1419126231_1103
A theater has $7$ actors, each assigned a unique costume. After a power outage, the costumes are returned such that no actor gets their own. In how many ways can this happen? Multiply that number by $73589$, then find the remainder when the result is divided by $77738$.
3,816
COMB
null
COUNT
sympy
SUM_GEOM
[ "SUM_GEOM" ]
04214c
comb_count_derangements_v1
null
3
null
[ "SUM_GEOM" ]
1
0.001
2026-02-25T10:38:04.445507Z
null
a79d42
548e84
narrative
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 171, "completion_tokens": 1014 }, "timestamp": "2026-03-31T04:17:42.506Z", "answer": 3816 }, { "i...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "SUM_GEOM", "status": "ok" }, { "lemma": "V8", "status": "no" }, { "lemma": "V8_SUM", "status": "no" ...
{ "lo": -5.37, "mid": 0.23, "hi": 5.22 }