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089712
comb_count_permutations_fixed_v1_1742523217_2994
Let $n$ be the number of integers $t$ such that there exist integers $a$ and $b$ satisfying $1 \leq a \leq 2$, $1 \leq b \leq 4$, $5 \leq t \leq 14$, and $t = 3a + 2b$. Let $k = 5$. Define $\text{result} = \binom{n}{k} \cdot !(n - k)$, where $!m$ denotes the number of derangements of $m$ elements. Let $Q$ be the remain...
46,975
graphs = [ Graph( let={ "_n": Const(67049), "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=Va...
COMB
null
COUNT
sympy
LIN_FORM
[ "LIN_FORM" ]
7b2633
comb_count_permutations_fixed_v1
null
5
0
[ "LIN_FORM" ]
1
0.002
2026-02-08T05:29:43.336965Z
{ "verified": true, "answer": 46975, "timestamp": "2026-02-08T05:29:43.339418Z" }
577399
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 261, "completion_tokens": 1526 }, "timestamp": "2026-02-24T03:39:27.412Z", "answer": 46975 }, { "...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "V8_SUM", "st...
{ "lo": -2.46, "mid": 0.47, "hi": 3.59 }
0b27e2
nt_sum_divisors_mod_v1_1125832087_579
Let $s$ be the minimum value of $x + y$ where $x$ and $y$ are positive integers such that $xy = 705600$. Let $n$ be the sum of $\phi(d)$ over all positive divisors $d$ of $s$, where $\phi$ denotes Euler's totient function. Let $\sigma(n)$ be the sum of all positive divisors of $n$. Compute the remainder when $\sigma(n)...
5,952
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")), Const(705600)))), expr=Sum(Var("x"), Var("y")))), "n": SumOverDiv...
NT
null
COMPUTE
sympy
B3
[ "B3/K3" ]
4a4ef2
nt_sum_divisors_mod_v1
null
6
0
[ "B3", "K3" ]
2
0.003
2026-02-08T03:09:25.157398Z
{ "verified": true, "answer": 5952, "timestamp": "2026-02-08T03:09:25.160426Z" }
2da422
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": 1740 }, "timestamp": "2026-02-10T13:14:31.451Z", "answer": 5952 }, { "i...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "K3", "status": "ok_later" }, { "lemma": "L3b", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "MAX_VAL", "status": "no" }, { "lemma": "V3", "status": "no" }, { "lemma"...
{ "lo": -3.46, "mid": 1.03, "hi": 5.49 }
8ba33b
algebra_vieta_sum_v1_458359167_4830
Let $s$ be the minimum value of $x + y$ over all positive integers $x$ and $y$ such that $xy = 2916$. Let $R$ be the set of all positive integers $x$ such that $$ x^3 - 19x^2 + s \cdot x - 180 = 0. $$ Compute the product of all elements in $R$.
180
graphs = [ Graph( let={ "_n": Const(3), "result": ProductOverSet(set=SolutionsSet(var=Var(name='x'), condition=Eq(left=Sum(Pow(base=Var(name='x'), exp=Ref(name='_n')), Mul(Const(value=-19), Pow(base=Var(name='x'), exp=Const(value=2))), Mul(MinOverSet(set=MapOverSet(set=SolutionsSet(v...
NT
null
COMPUTE
sympy
B3
[ "B3" ]
0cd20d
algebra_vieta_sum_v1
null
7
0
[ "B3" ]
1
0.013
2026-02-08T12:05:43.202089Z
{ "verified": true, "answer": 180, "timestamp": "2026-02-08T12:05:43.215030Z" }
8402a9
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": 1183 }, "timestamp": "2026-02-14T22:17:37.292Z", "answer": 180 }, { ...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "K14", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, { "lemma": "V8_SUM", "status": "no" } ]
{ "lo": -5.14, "mid": 0.32, "hi": 6.51 }
adb5d7
modular_mod_compute_v1_784195855_2571
Let $a = 19881$. Consider the set of all ordered pairs $(x,y)$ of positive integers such that $xy = 19749136$. Let $m$ be the minimum value of $x + y$ over all such pairs. Compute the remainder when $a$ is divided by $m$.
2,105
graphs = [ Graph( let={ "a": Const(19881), "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")), Const(19749136)))), expr=Sum(Var("x"), Var("y")...
NT
null
COMPUTE
sympy
B3
[ "B3" ]
0cd20d
modular_mod_compute_v1
null
4
0
[ "B3" ]
1
0.006
2026-02-08T05:52:39.322485Z
{ "verified": true, "answer": 2105, "timestamp": "2026-02-08T05:52:39.328745Z" }
e4da72
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 111, "completion_tokens": 720 }, "timestamp": "2026-02-12T16:06:53.871Z", "answer": 2105 }, { ...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, { "lemma": "MOD_MUL", "status": "no" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" }, { "lemma": "V1", "status": "no...
{ "lo": -3.51, "mid": 1.33, "hi": 6.56 }
cddb3e
comb_sum_binomial_row_v1_153355830_392
Let $A$ be the number of unordered pairs of coprime positive integers $p, q$ with $p < q$ and $pq = 216$. Let $B$ be the number of integers $t$ such that $7 \leq t \leq 27$ and $t = 4a + 3b$ for some integers $a, b$ with $1 \leq a \leq 3$ and $1 \leq b \leq 5$. Compute $A^B$.
32,768
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=216)), Eq(left=GCD(a=Var(name='p'), b=Var(name='q')), right=Const(value=1)),...
NT
null
SUM
sympy
COPRIME_PAIRS
[ "COPRIME_PAIRS/LIN_FORM" ]
a1eac8
comb_sum_binomial_row_v1
null
4
0
[ "COPRIME_PAIRS", "LIN_FORM" ]
2
0.002
2026-02-08T03:03:49.802553Z
{ "verified": true, "answer": 32768, "timestamp": "2026-02-08T03:03:49.804744Z" }
ced8bb
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 222, "completion_tokens": 1729 }, "timestamp": "2026-02-10T12:38:57.561Z", "answer": 32768 }, { "...
1
[ { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok_later" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, { "lemma": "POLY_PAD...
{ "lo": -3.47, "mid": 0.81, "hi": 4.81 }
9319dd
nt_max_prime_below_v1_677425708_2022
Let $ A $ be the set of all positive integers $ p $ for which there exists a positive integer $ q > p $ such that $ p \cdot q = 18 $ and $ \gcd(p, q) = 1 $. Let $ s $ be the number of elements in $ A $. Determine the largest prime number $ n $ such that $ s \leq n \leq 14161 $.
14,159
graphs = [ Graph( let={ "upper": Const(14161), "result": MaxOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), CountOverSet(set=SolutionsSet(var=Var("p"), condition=And(IsPositive(arg=Var(name='p')), Exists(var=Var(name='q'), condition=And(Eq(left=Mul(Var(name='p'), V...
NT
null
EXTREMUM
sympy
COPRIME_PAIRS
[ "COPRIME_PAIRS" ]
2bb3aa
nt_max_prime_below_v1
null
4
0
[ "COPRIME_PAIRS" ]
1
0.321
2026-02-08T04:43:08.318743Z
{ "verified": true, "answer": 14159, "timestamp": "2026-02-08T04:43:08.639778Z" }
e691b7
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 197, "completion_tokens": 1735 }, "timestamp": "2026-02-11T07:24:42.043Z", "answer": 14159 }, { ...
1
[ { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "K18", "status": "no" }, { "lemma": "MAX_VAL", "status": "no" }, { "lemma": "V1", "status": "no" }, { "lemma": "V3", "status": "no" }, { "lemma": "V8", "status": "no" } ]
{ "lo": -7.01, "mid": -4.68, "hi": -2.15 }
a30a45
nt_sum_gcd_range_mod_v1_48377204_1620
Let $N$ be the number of positive integers $n$ such that $1 \leq n \leq 34385$ and $$ n \equiv \left\lfloor \frac{n}{2} \right\rfloor \pmod{11}. $$ Let $k = \sum_{k_1=1}^{15} k_1$. Define $$ S = \sum_{n_1=1}^{N} \gcd(n_1, k). $$ Let $M = 11633$, and let $r$ be the remainder when $S$ is divided by $M$. Compute the r...
42,583
graphs = [ Graph( let={ "_n": Const(15), "N": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), Const(34385)), Congruent(a=Var(name='n'), b=Floor(arg=Div(left=Var(name='n'), right=Const(value=2))), modulus=Const(value=11))))), ...
NT
null
COMPUTE
sympy
SUM_ARITHMETIC
[ "SUM_ARITHMETIC", "L3C" ]
22c5b7
nt_sum_gcd_range_mod_v1
null
6
0
[ "L3C", "SUM_ARITHMETIC" ]
2
0.612
2026-02-08T16:15:49.289968Z
{ "verified": true, "answer": 42583, "timestamp": "2026-02-08T16:15:49.901617Z" }
f3854b
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 188, "completion_tokens": 2768 }, "timestamp": "2026-02-17T00:50:59.565Z", "answer": 42583 }, ...
1
[ { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "K14", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "L3C", "status": "ok" }, { "lemma": "LTE_DIFF_P2", "status": "no" ...
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
f1859e
comb_sum_binomial_row_v1_784195855_2340
Let $m = 245$. Let $d$ be the smallest divisor of $m$ that is at least $2$. Define $$ n = \sum_{k=1}^{5} \phi(k) \left\lfloor \frac{d}{k} \right\rfloor, $$ where $\phi$ denotes Euler's totient function. Compute $2^n$.
32,768
graphs = [ Graph( let={ "_m": Const(245), "_n": MinOverSet(set=SolutionsSet(var=Var("d"), condition=And(Geq(Var("d"), Const(2)), Divides(divisor=Var("d"), dividend=Ref("_m"))))), "n": Summation(var="k", start=Const(1), end=Const(5), expr=Mul(EulerPhi(n=Var("k")), Floor(Di...
NT
null
SUM
sympy
MIN_PRIME_FACTOR
[ "MIN_PRIME_FACTOR/K2" ]
352a97
comb_sum_binomial_row_v1
null
5
0
[ "K2", "MIN_PRIME_FACTOR" ]
2
0.002
2026-02-08T05:41:50.570958Z
{ "verified": true, "answer": 32768, "timestamp": "2026-02-08T05:41:50.573065Z" }
e7d44e
CC BY 4.0
[ { "id": 8, "model": "mathstral", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 144, "completion_tokens": 561 }, "timestamp": "2026-02-11T23:01:02.273Z", "answer": 32768 }, { "id": 11, ...
2
[ { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "K15", "status": "no" }, { "lemma": "K2", "status": "ok_later" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "MIN_PRIME_FACTOR", "s...
{ "lo": -9.14, "mid": -6.05, "hi": -3.73 }
7bb4c8
nt_count_digit_sum_v1_2051736721_2440
Let $S$ be the set of all positive integers $p$ for which there exists a positive integer $q$ such that $pq = 323400$, $\gcd(p, q) = 1$, and $p < q$. Let $T$ be the number of elements in $S$. Now let $N$ be the number of positive integers $n \leq 9999$ such that the sum of the digits of $n$ equals $T$. Compute the Bell...
203
graphs = [ Graph( let={ "upper": Const(9999), "target_sum": 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=323400)), Eq(left=GCD(a=Var(name='...
NT
COMB
COUNT
sympy
LIN_FORM
[ "COPRIME_PAIRS" ]
2bb3aa
nt_count_digit_sum_v1
null
5
0
[ "COPRIME_PAIRS", "LIN_FORM" ]
2
12.55
2026-02-08T16:40:36.635520Z
{ "verified": true, "answer": 203, "timestamp": "2026-02-08T16:40:49.185359Z" }
ffb3d9
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 167, "completion_tokens": 2319 }, "timestamp": "2026-02-17T10:32:10.671Z", "answer": 203 }, { ...
1
[ { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "K18", "status": "no" }, { "lemma": "L3c", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" }, { "lemma": "V5", "status": "no" } ]
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
4dd571
comb_sum_binomial_row_v1_717093673_2819
Let $n$ be the number of integers $t$ with $7 \leq t \leq 25$ for which there exist integers $a$ and $b$ such that $1 \leq a \leq 5$, $1 \leq b \leq 3$, and $t = 2a + 5b$. Let $w = \sum_{k=0}^{0} (-1)^k \binom{0}{k}$ and $u = \sum_{k_1=0}^{0} (-1)^{k_1} \binom{0}{k_1}$. Define $N = n \cdot w \cdot u$. Compute $2^N$.
32,768
graphs = [ Graph( let={ "_n": Const(2), "n2": Const(0), "w": Summation(var="k", start=Sub(Binom(n=Const(16), k=Const(16)), Const(1)), end=Ref("n2"), expr=Mul(Pow(Const(-1), Var("k")), Binom(n=Ref("n2"), k=Var("k")))), "n1": Const(0), "u": Summation...
COMB
null
SUM
sympy
LIN_FORM
[ "LIN_FORM/BINOMIAL_ALTERNATING", "ZERO_BINOM_N" ]
3ea266
comb_sum_binomial_row_v1
null
5
2
[ "BINOMIAL_ALTERNATING", "LIN_FORM", "ZERO_BINOM_N" ]
3
0.005
2026-02-08T17:13:01.922712Z
{ "verified": true, "answer": 32768, "timestamp": "2026-02-08T17:13:01.927217Z" }
76ee0f
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": 951 }, "timestamp": "2026-02-17T21:53:12.741Z", "answer": 32768 }, {...
1
[ { "lemma": "BINOMIAL_ALTERNATING", "status": "ok_later" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", ...
{ "lo": -4.1, "mid": -1.76, "hi": 1.32 }
8aaa87
antilemma_k3_v1_677425708_2301
Let $x$ be the sum of $\phi(d)$ over all positive divisors $d$ of $4375$, where $\phi$ denotes Euler's totient function. Compute the value of $x$.
4,375
graphs = [ Graph( let={ "x": SumOverDivisors(n=Const(value=4375), 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
2026-02-08T04:58:48.101389Z
{ "verified": true, "answer": 4375, "timestamp": "2026-02-08T04:58:48.101759Z" }
0532d3
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 153, "completion_tokens": 452 }, "timestamp": "2026-02-11T22:36:18.607Z", "answer": 4375 }, { "i...
1
[ { "lemma": "DS2", "status": "no" }, { "lemma": "K13", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "K3", "status": "ok" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "V5", "status": "no" } ]
{ "lo": -3.42, "mid": 1.86, "hi": 7.55 }
ce50e4
nt_sum_gcd_range_mod_v1_1520064083_822
Let $N = 4356$ and $M = 11497$. Define $k$ to be the minimum value of $x + y$ over all ordered pairs of positive integers $(x, y)$ such that $xy = 20736$. Let $s = \sum_{n=1}^{N} \gcd(n, k)$. Define $c$ to be the minimum value of $x + y$ over all ordered pairs of positive integers $(x, y)$ such that $xy = 7817616$. Fin...
57,208
graphs = [ Graph( let={ "_n": Const(87617), "N": Const(4356), "k": 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(20736)))),...
NT
null
COMPUTE
sympy
B3
[ "B3" ]
e0298c
nt_sum_gcd_range_mod_v1
affine_mod
5
0
[ "B3" ]
1
0.202
2026-02-08T03:37:28.860306Z
{ "verified": true, "answer": 57208, "timestamp": "2026-02-08T03:37:29.062049Z" }
c47849
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 243, "completion_tokens": 5011 }, "timestamp": "2026-02-10T15:07:09.433Z", "answer": 57208 }, { "...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "K13", "status": "no" }, { "lemma": "L3c", "status": "no" }, { "lemma": "V8_SUM", "status": "no" ...
{ "lo": -3.46, "mid": 1.03, "hi": 5.49 }
4b4282
nt_num_divisors_compute_v1_784195855_2458
Let $n$ be the sum of all real solutions $x$ to the equation $$ x^2 - 2020x - 32576 = 0. $$ Compute the number of positive divisors of $n$.
12
graphs = [ Graph( let={ "_n": Const(2), "n": SumOverSet(set=SolutionsSet(var=Var("x"), condition=Eq(Sum(Pow(Var("x"), Ref("_n")), Mul(Const(-2020), Var("x")), Const(-32576)), Const(0)))), "result": NumDivisors(n=Ref("n")), }, goal=Ref("result"), ) ]
NT
null
COMPUTE
sympy
VIETA_SUM
[ "VIETA_SUM" ]
b33a7a
nt_num_divisors_compute_v1
null
2
0
[ "VIETA_SUM" ]
1
0.002
2026-02-08T05:45:12.491572Z
{ "verified": true, "answer": 12, "timestamp": "2026-02-08T05:45:12.493483Z" }
be9d6c
CC BY 4.0
[ { "id": 8, "model": "mathstral", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 109, "completion_tokens": 747 }, "timestamp": "2026-02-11T23:09:11.978Z", "answer": 12 }, { "id": 11, ...
2
[ { "lemma": "K14", "status": "no" }, { "lemma": "L3c", "status": "no" }, { "lemma": "MAX_VAL", "status": "no" }, { "lemma": "V5", "status": "no" }, { "lemma": "V8_SUM", "status": "no" }, { "lemma": "VIETA_SUM", "status": "ok" } ]
{ "lo": -10, "mid": -7.73, "hi": -5.46 }
3b1ba8
antilemma_sum_equals_v1_1520064083_4456
Let $\mathcal{S}$ be the set of all ordered pairs $(i, j)$ of integers such that $1 \leq i \leq 97$, $1 \leq j \leq 98$, and $i + j = 98$. Let $x$ be the number of elements in $\mathcal{S}$. Compute the remainder when $44121 \cdot x$ is divided by $76826$.
54,307
graphs = [ Graph( let={ "x": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var("i"), Var("j")]), condition=Eq(Sum(Var("i"), Var("j")), Const(98)), domain=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(97)), right=IntegerRange(start=Const(1), end=Const(98))))), "Q": ...
COMB
GEOM
COMPUTE
sympy
COUNT_SUM_EQUALS
[ "COUNT_SUM_EQUALS" ]
75ab0f
antilemma_sum_equals_v1
null
3
0
[ "COUNT_SUM_EQUALS" ]
1
0.004
2026-02-08T06:17:26.805426Z
{ "verified": true, "answer": 54307, "timestamp": "2026-02-08T06:17:26.809798Z" }
ba6208
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 202, "completion_tokens": 1563 }, "timestamp": "2026-02-24T05:46:49.253Z", "answer": 54307 }, { "...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "ok" }, { "lemma": "V8", ...
{ "lo": -3.84, "mid": -1.69, "hi": 1.09 }
ad3110_n
alg_qf_psd_orbit_v1_1218484723_3420
An architect designs square rooms with side lengths $a$ and $b$ (in meters), where $a \le b$ and both are positive integers up to 195. The combined area of two such squares, scaled by a factor of 25, must equal 616250 square meters. How many distinct room-pair designs satisfy this condition?
6
ALG
null
COUNT
sympy
POLY4_COUNT
[ "LIN_FORM/POLY_ORBIT_HENSEL", "B1/POLY_ORBIT_HENSEL" ]
5d1f37
alg_qf_psd_orbit_v1
null
2
null
[ "B1", "LIN_FORM", "POLY4_COUNT", "POLY_ORBIT_HENSEL" ]
4
1.198
2026-02-25T05:07:49.932115Z
null
4e80b1
ad3110
narrative
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 181, "completion_tokens": 6460 }, "timestamp": "2026-03-30T20:03:31.641Z", "answer": 6 }, { "id":...
1
[ { "lemma": "B1", "status": "ok" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "POLY_ORBIT_HENSEL", "status": "ok_later" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" } ]
{ "lo": -5.37, "mid": 0.23, "hi": 5.22 }
8a52d9
nt_max_prime_below_v1_1978505735_2090
Let $P$ be the set of all positive integers $p$ for which there exists a positive integer $q$ such that $pq = 108$, $\gcd(p, q) = 1$, and $p < q$. Let $m$ be the number of elements in $P$. Let $S$ be the set of all prime numbers $n$ such that $m \leq n \leq 10883$. Determine the value of the largest element in $S$.
10,883
graphs = [ Graph( let={ "upper": Const(10883), "result": MaxOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), CountOverSet(set=SolutionsSet(var=Var("p"), condition=And(IsPositive(arg=Var(name='p')), Exists(var=Var(name='q'), condition=And(Eq(left=Mul(Var(name='p'), V...
NT
null
EXTREMUM
sympy
COPRIME_PAIRS
[ "COPRIME_PAIRS" ]
2bb3aa
nt_max_prime_below_v1
null
5
0
[ "COPRIME_PAIRS" ]
1
0.28
2026-02-08T16:39:12.714333Z
{ "verified": true, "answer": 10883, "timestamp": "2026-02-08T16:39:12.994115Z" }
8c0cf2
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": 1577 }, "timestamp": "2026-02-17T08:54:10.272Z", "answer": 10883 }, ...
1
[ { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "K13", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "L3b", "status": "no" }, { "lemma": "MOD_ADD", "status": "no" }, { "lemma": "V8", "status": "no" } ]
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
664f46
nt_min_coprime_above_v1_1978505735_2296
Let $n$ be the number of integers $n'$ such that $1 \leq n' \leq 184$ and the sum of the decimal digits of $n'$ is even. Let $s = 12321$ and $u = 12367$. Let $r$ be the smallest integer $n_1$ such that $s < n_1 \leq u$ and $\gcd(n_1, 36) = 1$. Let $p$ be the largest prime number at most $n$. Compute the remainder when ...
41,866
graphs = [ Graph( let={ "_m": Const(2), "_n": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), Const(184)), Eq(Mod(value=DigitSum(Var("n")), modulus=Ref("_m")), Const(0))))), "start": Const(12321), "upper": Const...
NT
null
EXTREMUM
sympy
L3B
[ "L3B/MAX_PRIME_BELOW" ]
ec756f
nt_min_coprime_above_v1
negation_mod
5
0
[ "L3B", "MAX_PRIME_BELOW" ]
2
0.012
2026-02-08T16:49:01.783373Z
{ "verified": true, "answer": 41866, "timestamp": "2026-02-08T16:49:01.795458Z" }
1d4d3d
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 173, "completion_tokens": 1907 }, "timestamp": "2026-02-17T12:36:45.253Z", "answer": 41866 }, ...
1
[ { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "L3B", "status": "ok" }, { "lemma": "L3b", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok_later" }, { "lemma": "V8", "status": "no" ...
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
e6a00f
antilemma_sum_equals_v1_458359167_3183
Let $n = 6$. Determine the number of ordered pairs $(i, j)$ of positive integers such that $1 \leq i \leq 5$, $1 \leq j \leq 6$, and $i + j = n$.
5
graphs = [ Graph( let={ "_n": Const(6), "x": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var("i"), Var("j")]), condition=Eq(Sum(Var("i"), Var("j")), Ref("_n")), domain=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(5)), right=IntegerRange(start=Const(1), end=Const...
COMB
GEOM
COMPUTE
sympy
COUNT_SUM_EQUALS
[ "COUNT_SUM_EQUALS" ]
75ab0f
antilemma_sum_equals_v1
null
3
0
[ "COUNT_SUM_EQUALS" ]
1
0.012
2026-02-08T07:00:55.764474Z
{ "verified": true, "answer": 5, "timestamp": "2026-02-08T07:00:55.776200Z" }
6e6e23
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 165, "completion_tokens": 174 }, "timestamp": "2026-02-24T07:27:40.906Z", "answer": 5 }, { "id": ...
2
[ { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "ok" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", "status": "no" }, { "lemma": "V8_SUM", "status":...
{ "lo": -10, "mid": -7.79, "hi": -5.57 }
02b42d
antilemma_sum_equals_v1_1915831931_2861
Let $n$ be the number of ordered pairs $(i, j)$ such that $1 \leq i \leq 8$ and $1 \leq j \leq 12$. Compute the number of ordered pairs $(i, j)$ of positive integers such that $1 \leq i \leq 94$, $1 \leq j \leq 95$, and $i + j = n$.
94
graphs = [ Graph( let={ "_n": CountOverSet(set=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(8)), right=IntegerRange(start=Const(1), end=Const(12)))), "x": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var("i"), Var("j")]), condition=Eq(Sum(Var("i"), Var("j")), Ref...
COMB
GEOM
COMPUTE
sympy
LIN_FORM
[ "COUNT_CARTESIAN/COUNT_SUM_EQUALS", "COUNT_SUM_EQUALS" ]
8bee9e
antilemma_sum_equals_v1
null
3
0
[ "COUNT_CARTESIAN", "COUNT_SUM_EQUALS", "LIN_FORM" ]
3
0.125
2026-02-08T17:11:46.658921Z
{ "verified": true, "answer": 94, "timestamp": "2026-02-08T17:11:46.784184Z" }
c09e33
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 129, "completion_tokens": 727 }, "timestamp": "2026-02-17T22:28:39.314Z", "answer": 94 }, { ...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_CARTESIAN", "status": "ok" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "ok" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", "status": "no...
{ "lo": -8, "mid": -4.75, "hi": -2.28 }
4a6ec4
nt_lcm_compute_v1_151522320_745
Let $A$ be the set of all integers $t$ such that $7 \leq t \leq 894$ and there exist positive integers $a$ and $b$ with $1 \leq a \leq 97$, $1 \leq b \leq 140$, and $t = 2a + 5b$. Let $a = |A|$. Compute the remainder when $24025 - \text{lcm}(a, 2522)$ is divided by $65505$.
3,782
graphs = [ Graph( let={ "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=97)), Geq(left=Var(name='b'), right=Const(value=...
NT
null
COMPUTE
sympy
LIN_FORM
[ "LIN_FORM" ]
7b2633
nt_lcm_compute_v1
null
6
0
[ "LIN_FORM" ]
1
0.002
2026-02-08T03:29:36.186212Z
{ "verified": true, "answer": 3782, "timestamp": "2026-02-08T03:29:36.187768Z" }
5e10da
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 221, "completion_tokens": 3881 }, "timestamp": "2026-02-10T13:36:32.376Z", "answer": 20364 }, { ...
1
[ { "lemma": "K16", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" }, { "lemma": "V1", "status": "no" }, { "lemma": "V3", "status": "no" }, { "lemma": "V8_SUM", "status": "no" } ]
{ "lo": 1.96, "mid": 5.23, "hi": 8.52 }
3b09ce
antilemma_k2_v1_784195855_5397
Let $m = 2$ and $n = 372$. Let $S$ be the set of all integers $x$ such that $x^2 - 372x + 17696 = 0$. Let $k_{\text{sum}}$ be the sum of all positive integers $k$ from 1 to the sum of the elements in $S$, inclusive. Define $$ x = \sum_{k=1}^{k_{\text{sum}}} \phi(k) \left\lfloor \frac{1}{k} \sum_{d \mid 372} \phi(d) \ri...
37,520
graphs = [ Graph( let={ "_m": Const(2), "_n": Const(372), "x": Summation(var="k", start=Const(1), end=SumOverSet(set=SolutionsSet(var=Var("x"), condition=Eq(Sum(Pow(Var("x"), Ref("_m")), Mul(Const(-372), Var("x")), Const(17696)), Const(0)))), expr=Mul(EulerPhi(n=Var("k"))...
NT
COMB
COMPUTE
sympy
K13
[ "VIETA_SUM/K2", "K3/K2", "K2" ]
4108ea
antilemma_k2_v1
null
7
0
[ "K13", "K2", "K3", "VIETA_SUM" ]
4
0.006
2026-02-08T07:52:34.063820Z
{ "verified": true, "answer": 37520, "timestamp": "2026-02-08T07:52:34.069691Z" }
403dc4
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 211, "completion_tokens": 1549 }, "timestamp": "2026-02-13T13:17:15.008Z", "answer": 37520 }, ...
1
[ { "lemma": "K14", "status": "no" }, { "lemma": "K15", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "K2", "status": "ok" }, { "lemma": "K3", "status": "ok" }, { "lemma": "L3b", "status": "no" }, { "lemma": "VAL_SUM_EQ", ...
{ "lo": -3.44, "mid": 1.34, "hi": 6.56 }
0a7aea
comb_count_derangements_v1_458359167_3158
Let $n$ be the largest prime number not exceeding $7$. Let $r = !n$, the number of derangements of $n$ elements. Let $c$ be the maximum value of $xy$ over all pairs of positive integers $(x, y)$ such that $x + y = 14$. Compute the value of $$\sum_{i=0}^{d-1} \left( \text{digit}_i(r) \cdot (i+1)^2 \right) + c,$$ where $...
161
graphs = [ Graph( let={ "_m": Const(7), "_n": Const(2), "n": MaxOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(2)), Leq(Var("n"), Ref("_m")), IsPrime(Var("n"))))), "result": Subfactorial(arg=Ref(name='n')), "_c": MaxOverSe...
NT
COMB
COUNT
sympy
B1
[ "B1", "MAX_PRIME_BELOW" ]
913a2e
comb_count_derangements_v1
digits_weighted_mod
5
0
[ "B1", "MAX_PRIME_BELOW" ]
2
0.003
2026-02-08T07:00:06.470226Z
{ "verified": true, "answer": 161, "timestamp": "2026-02-08T07:00:06.473341Z" }
bf6a78
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": 816 }, "timestamp": "2026-02-13T07:05:38.506Z", "answer": 161 }, { ...
1
[ { "lemma": "B1", "status": "ok" }, { "lemma": "K14", "status": "no" }, { "lemma": "L3c", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, { ...
{ "lo": -7.01, "mid": -4.68, "hi": -2.15 }
e8d8e9
modular_mod_compute_v1_865884756_6767
Let $m$ be the largest prime number less than or equal to $1213$. Compute the remainder when $512$ is divided by $m$.
512
graphs = [ Graph( let={ "_n": Const(1213), "a": Const(512), "m": MaxOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(2)), Leq(Var("n"), Ref("_n")), IsPrime(Var("n"))))), "result": Mod(value=Ref("a"), modulus=Ref("m")), }, ...
NT
null
COMPUTE
sympy
MAX_PRIME_BELOW
[ "MAX_PRIME_BELOW" ]
dc3ad3
modular_mod_compute_v1
null
2
0
[ "MAX_PRIME_BELOW" ]
1
0.002
2026-02-08T19:22:55.674087Z
{ "verified": true, "answer": 512, "timestamp": "2026-02-08T19:22:55.675998Z" }
5da66d
CC BY 4.0
[ { "id": 8, "model": "mathstral", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 90, "completion_tokens": 383 }, "timestamp": "2026-02-16T18:38:10.550Z", "answer": 512 }, { "id": 11, ...
2
[ { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok" }, { "lemma": "MOD_MUL", "status": "no" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" }, { "lemma": "V8",...
{ "lo": -10, "mid": -7.27, "hi": -4.54 }
764f0f
comb_factorial_compute_v1_153355830_2005
Let $S$ be the set of all ordered pairs $(x, y)$ of positive integers such that $xy = 16$. Let $T$ be the set of all values $x + y$ where $(x, y) \in S$. Let $m$ be the minimum element of $T$. Let $n$ be the largest prime number satisfying $2 \leq n \leq m$. Compute $n!$.
5,040
graphs = [ Graph( let={ "_n": Const(2), "n": MaxOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Ref("_n")), Leq(Var("n"), MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var("x"), Var("y")]), condition=And(IsPositive(arg=Var(name='x')), IsPositive(ar...
NT
null
COMPUTE
sympy
B3
[ "B3/MAX_PRIME_BELOW" ]
f253c0
comb_factorial_compute_v1
null
3
0
[ "B3", "MAX_PRIME_BELOW" ]
2
0.002
2026-02-08T06:50:55.488388Z
{ "verified": true, "answer": 5040, "timestamp": "2026-02-08T06:50:55.489941Z" }
2d626b
CC BY 4.0
[ { "id": 8, "model": "mathstral", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 145, "completion_tokens": 469 }, "timestamp": "2026-02-15T17:47:25.076Z", "answer": 5040 }, { "id": 11, ...
2
[ { "lemma": "B3", "status": "ok" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "K18", "status": "no" }, { "lemma": "L3b", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok_later" }, { "lemma": "MAX_VAL", "status": "no...
{ "lo": -10, "mid": -7.73, "hi": -5.46 }
307802
modular_count_residue_v1_153355830_550
Let $n$ be a positive integer such that $1 \leq n \leq 50625$ and $n \equiv 1 \pmod{3}$. Compute the number of such integers $n$. Let $c$ be the largest prime number less than or equal to 5004. Find the remainder when $\left( (\text{number of such } n) \bmod{199} \right) + c \cdot \left( (\text{number of such } n) \...
41,958
graphs = [ Graph( let={ "upper": Const(50625), "m": Const(3), "r": Const(1), "result": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), Ref("upper")), Eq(Mod(value=Var("n"), modulus=Ref("m")), Ref("r"))))), ...
NT
null
COUNT
sympy
MAX_PRIME_BELOW
[ "MAX_PRIME_BELOW" ]
08a653
modular_count_residue_v1
two_moduli
4
0
[ "MAX_PRIME_BELOW" ]
1
2.828
2026-02-08T03:09:33.536452Z
{ "verified": true, "answer": 41958, "timestamp": "2026-02-08T03:09:36.363975Z" }
5d2192
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": 2144 }, "timestamp": "2026-02-10T15:14:06.895Z", "answer": 41958 }, { ...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "L3c", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok" }, { "lemma": "MOD_MUL", "status": "no" }, { "lemma": "V3", "status": "no" } ]
{ "lo": -3.42, "mid": 1.86, "hi": 7.55 }
1d74e4
diophantine_fbi2_count_v1_124444284_3042
Let $k = 840$. Determine the number of positive integers $d$ such that $4 \leq d \leq 114$, $d$ divides $k$, and $3 \leq \frac{k}{d} \leq 113$. Call this number $r$. Let $m = 1225$. Consider the set of all ordered pairs $(x, y)$ of positive integers such that $xy = m$. Let $c$ be the minimum value of $x + y$ over all ...
1,276
graphs = [ Graph( let={ "_m": Const(1225), "_n": Const(113), "k": Const(840), "result": CountOverSet(set=SolutionsSet(var=Var("d"), condition=And(Geq(Var("d"), Const(4)), Leq(Var("d"), Const(114)), Divides(divisor=Var("d"), dividend=Ref("k")), Geq(Div(Ref("k")...
NT
null
COUNT
sympy
MAX_VAL
[ "MAX_VAL", "B3" ]
82a41b
diophantine_fbi2_count_v1
quadratic_mod
5
0
[ "B3", "MAX_VAL" ]
2
0.039
2026-02-08T05:09:57.556258Z
{ "verified": true, "answer": 1276, "timestamp": "2026-02-08T05:09:57.594831Z" }
2a7997
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 263, "completion_tokens": 2865 }, "timestamp": "2026-02-11T23:06:53.208Z", "answer": 1276 }, { "i...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "K14", "status": "no" }, { "lemma": "K15", "status": "no" }, { "lemma": "K16", "status": "same_pattern_wrong" }, { "lemma": "K5", "status": "no" }, ...
{ "lo": -3.45, "mid": 1.15, "hi": 6.18 }
fb9824
antilemma_v1_legendre_168721529_655
Let $m=4$ and $N=29764$. Let $p$ be the greatest prime number between $2$ and $12$, inclusive. Let $x$ be the largest integer $k$ such that $p^k$ divides $74756!$. Let $T$ be the set of all integers $t$ such that there exist integers $a$ and $b$ with $1\le a\le 5$, $1\le b\le 3$, $7\le t\le 25$, and $$t = 2a + 5b.$$ L...
54,624
graphs = [ Graph( let={ "_c": Const(58129), "_m": Const(4), "_n": Const(29764), "x": MaxKDivides(target=Factorial(Const(74756)), base=MaxOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(2)), Leq(Var("n"), Const(12)), IsPrime(Var("n"))))...
NT
null
COMPUTE
sympy
LIN_FORM
[ "LIN_FORM/C5", "MAX_PRIME_BELOW/V1", "V1" ]
07e81d
antilemma_v1_legendre
negation_mod
7
0
[ "C5", "LIN_FORM", "MAX_PRIME_BELOW", "V1" ]
4
0.005
2026-02-08T13:10:46.110185Z
{ "verified": true, "answer": 54624, "timestamp": "2026-02-08T13:10:46.115148Z" }
e58624
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 341, "completion_tokens": 2206 }, "timestamp": "2026-02-09T07:31:06.994Z", "answer": 54624 }, { "...
1
[ { "lemma": "C5", "status": "ok_later" }, { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "K5", "status": "same_pattern_wrong" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok" }, { "lemma": "MOD_FAC...
{ "lo": -1.99, "mid": 2.14, "hi": 6.33 }
561334
comb_sum_binomial_row_v1_349078426_1216
Let $n_0 = 21$. Let $S$ be the set of all positive integers $d$ such that $1 \leq d \leq 23$ and $d$ divides $667$. Let $M$ be the maximum element of $S$. Let $T$ be the set of all positive integers $n$ such that $1 \leq n \leq M$ and $\gcd(n, n_0) = 1$. Let $n$ be the number of elements in $T$. Compute $2^n$.
16,384
graphs = [ Graph( let={ "_n": Const(21), "n": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), MaxOverSet(set=SolutionsSet(var=Var("d"), condition=And(Geq(Var("d"), Const(1)), Leq(Var("d"), Const(23)), Divides(divisor=Var("d"), dividend...
NT
null
SUM
sympy
MAX_DIVISOR
[ "MAX_DIVISOR/C4" ]
ac1e32
comb_sum_binomial_row_v1
null
4
0
[ "C4", "MAX_DIVISOR" ]
2
0.002
2026-02-08T13:31:14.629931Z
{ "verified": true, "answer": 16384, "timestamp": "2026-02-08T13:31:14.631892Z" }
703374
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 158, "completion_tokens": 771 }, "timestamp": "2026-02-15T16:48:56.693Z", "answer": 16384 }, {...
1
[ { "lemma": "C4", "status": "ok_later" }, { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "DS2", "status": "no" }, { "lemma": "K15", "status": "no" }, { "lemma": "L3b", "status": "no" }, { "lemma": "L3c", "status": "no" }, { "lemma":...
{ "lo": -8.26, "mid": -5.01, "hi": -1.76 }
9809f6
comb_factorial_compute_v1_397696148_2105
Let $S$ be the set of all positive integers $p$ for which there exists a positive integer $q$ such that $p < q$, $\gcd(p, q) = 1$, and $p \cdot q = 24$. Let $m$ be the number of elements in $S$. Let $n$ be the largest prime number satisfying $m \le n \le 8$. Compute $n!$.
5,040
graphs = [ Graph( let={ "_n": Const(8), "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
comb_factorial_compute_v1
null
4
0
[ "COPRIME_PAIRS", "MAX_PRIME_BELOW" ]
2
0.002
2026-02-08T12:57:24.162188Z
{ "verified": true, "answer": 5040, "timestamp": "2026-02-08T12:57:24.164274Z" }
fa03ff
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 136, "completion_tokens": 653 }, "timestamp": "2026-02-15T07:43:31.353Z", "answer": 5040 }, { ...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok_later" }, { "lemma": "MAX_VAL", "sta...
{ "lo": -9.12, "mid": -6.02, "hi": -3.62 }
fa801e
algebra_quadratic_discriminant_v1_48377204_270
Let $ a = -9 $, $ b = 6 $, and $ m = 21 $. Define $ c = \frac{3}{m} \sum_{k=1}^{3} \sum_{j=1}^{7} k $. Compute the value of $ b^2 - 4ac $.
252
graphs = [ Graph( let={ "_m": Const(21), "_n": Const(4), "a": Const(-9), "b": Const(6), "c": Div(Mul(Const(3), SumOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var("k"), Var("_j")]), condition=Const(1), domain=CartesianProduct(left=In...
NT
null
COMPUTE
sympy
V1
[ "SUM_INDEPENDENT/SUM_ARITHMETIC" ]
87e6cf
algebra_quadratic_discriminant_v1
null
2
0
[ "SUM_ARITHMETIC", "SUM_INDEPENDENT", "V1" ]
3
0.009
2026-02-08T15:19:51.457408Z
{ "verified": true, "answer": 252, "timestamp": "2026-02-08T15:19:51.466888Z" }
4ebb54
CC BY 4.0
[ { "id": 8, "model": "mathstral", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 120, "completion_tokens": 265 }, "timestamp": "2026-02-16T05:22:44.601Z", "answer": 252 }, { "id": 11, ...
2
[ { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "K15", "status": "no" }, { "lemma": "MOD_ADD", "status": "no" }, { "lemma": "MOD_MUL", "status": "no" }, { "lemma": "SUM_ARITHMETIC", "status": "ok_later" }, { "lemma": "SUM_INDEPENDENT", ...
{ "lo": -10, "mid": -7.27, "hi": -4.54 }
a057df
antilemma_sum_equals_v1_677425708_1147
Let $n$ be the number of ordered pairs $(a, b)$ such that $1 \leq a \leq 7$ and $1 \leq b \leq 11$. Let $x$ be the number of ordered pairs $(i, j)$ of positive integers such that $i + j = n$, $1 \leq i \leq 75$, and $1 \leq j \leq 76$. Determine the value of $x$.
75
graphs = [ Graph( let={ "_n": CountOverSet(set=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(7)), right=IntegerRange(start=Const(1), end=Const(11)))), "x": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var("i"), Var("j")]), condition=Eq(Sum(Var("i"), Var("j")), Ref...
COMB
GEOM
COMPUTE
sympy
COUNT_SUM_EQUALS
[ "COUNT_CARTESIAN/COUNT_SUM_EQUALS", "COUNT_SUM_EQUALS" ]
8bee9e
antilemma_sum_equals_v1
null
3
0
[ "COUNT_CARTESIAN", "COUNT_SUM_EQUALS" ]
2
0.061
2026-02-08T04:01:12.396540Z
{ "verified": true, "answer": 75, "timestamp": "2026-02-08T04:01:12.457251Z" }
b42d9d
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 211, "completion_tokens": 3924 }, "timestamp": "2026-02-09T16:15:10.857Z", "answer": 75 }, { "id"...
1
[ { "lemma": "COUNT_CARTESIAN", "status": "ok" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "ok" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", ...
{ "lo": -5.97, "mid": -3.96, "hi": -1.93 }
cde006
comb_count_derangements_v1_717093673_2418
Let $n_1$ range over the positive integers from 1 to 8. Define $n$ to be the sum of all such $n_1$ for which $n_1$ is divisible by 8. Compute the number of derangements of $n$ elements, denoted $!n$.
14,833
graphs = [ Graph( let={ "_n": Const(8), "n": SumOverSet(set=SolutionsSet(var=Var("n1"), condition=And(Geq(Var("n1"), Const(1)), Leq(Var("n1"), Ref("_n")), Eq(Mod(value=Var("n1"), modulus=Const(8)), Const(0))))), "result": Subfactorial(arg=Ref(name='n')), }, ...
COMB
null
COUNT
sympy
SUM_DIVISIBLE
[ "SUM_DIVISIBLE" ]
02dbe3
comb_count_derangements_v1
null
3
0
[ "SUM_DIVISIBLE" ]
1
0.003
2026-02-08T16:50:04.928221Z
{ "verified": true, "answer": 14833, "timestamp": "2026-02-08T16:50:04.931329Z" }
fb9a21
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 111, "completion_tokens": 1147 }, "timestamp": "2026-02-17T12:29:15.554Z", "answer": 14833 }, ...
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": "SUM_DIVISIBLE", "status": "ok" }, { "lem...
{ "lo": -5.64, "mid": -3.17, "hi": -0.81 }
db51f2
geo_count_lattice_rect_v1_677425708_4111
Let $a = 81$ and $b = 29$. Define $R$ to be the number of lattice points $(x, y)$ such that $0 \leq x \leq a$ and $0 \leq y \leq b$. Let $r$ be the remainder when $|R|$ is divided by 11. Compute the $r$-th Bell number.
877
graphs = [ Graph( let={ "a": Const(81), "b": Const(29), "result": LatticePointsRect(a=Ref(name='a'), b=Ref(name='b')), "Q": Bell(Mod(value=Abs(arg=Ref(name='result')), modulus=Const(11))), }, goal=Ref("Q"), ) ]
GEOM
COMB
COUNT
sympy
[]
geo_count_lattice_rect_v1
null
3
0
null
null
0.001
2026-02-08T06:25:58.842542Z
{ "verified": true, "answer": 877, "timestamp": "2026-02-08T06:25:58.843049Z" }
eba370
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 194, "completion_tokens": 699 }, "timestamp": "2026-02-24T06:08:34.408Z", "answer": 877 }, { "id"...
1
[]
{ "lo": -4.93, "mid": -2.91, "hi": -0.71 }
1b8e9a
nt_sum_divisors_mod_v1_1918700295_1374
Let $n$ be the number of positive integers $k$ such that the smallest sum $x + y$ over all ordered pairs $(x, y)$ of positive integers with $xy = 225$ divides the $k$-th Fibonacci number, and $k \leq 302400$. Let $\sigma(n)$ denote the sum of all positive divisors of $n$. Compute the remainder when $\sigma(n)$ is divid...
8,975
graphs = [ Graph( let={ "_n": Const(225), "n": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), Const(302400)), Divides(divisor=MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var("x"), Var("y")]), condition=And(IsPositiv...
NT
null
COMPUTE
sympy
B3
[ "B3/COUNT_FIB_DIVISIBLE" ]
9b7a57
nt_sum_divisors_mod_v1
null
7
0
[ "B3", "COUNT_FIB_DIVISIBLE" ]
2
0.003
2026-02-08T05:48:31.131218Z
{ "verified": true, "answer": 8975, "timestamp": "2026-02-08T05:48:31.134637Z" }
fc0775
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": 1411 }, "timestamp": "2026-02-12T14:14:52.697Z", "answer": 8975 }, {...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "ok_later" }, { "lemma": "K18", "status": "no" }, { "lemma": "L3c", "status": "no" }, { "lemma": "LTE_DIFF", "status":...
{ "lo": -3.51, "mid": 1.33, "hi": 6.56 }
233d13
antilemma_k3_v1_458359167_3835
Let $m = 55191$ and $n = 39192$. Define $x$ to be the sum of $\phi(d)$ over all positive divisors $d$ of $n$. Let $A$ be the sum of $\phi(d)$ over all positive divisors $d$ of $7229$. Let $B$ be the sum of $\phi(d)$ over all positive divisors $d$ of $A$. Compute the remainder when $B \cdot x$ is divided by $m$.
23,565
graphs = [ Graph( let={ "_m": Const(55191), "_n": Const(39192), "x": SumOverDivisors(n=Ref(name='_n'), var='d', expr=EulerPhi(n=Var(name='d'))), "Q": Mod(value=Mul(SumOverDivisors(n=SumOverDivisors(n=Const(value=7229), var='d', expr=EulerPhi(n=Var(name='d'))),...
NT
COMB
COMPUTE
sympy
K3
[ "K3/K3", "K3" ]
574452
antilemma_k3_v1
affine_mod
4
0
[ "K3" ]
1
0.002
2026-02-08T11:23:18.934186Z
{ "verified": true, "answer": 23565, "timestamp": "2026-02-08T11:23:18.936259Z" }
a2ac8a
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 150, "completion_tokens": 6781 }, "timestamp": "2026-02-14T13:48:56.017Z", "answer": 23565 }, ...
1
[ { "lemma": "DS2", "status": "no" }, { "lemma": "K14", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "K3", "status": "ok" }, { "lemma": "MOD_MUL", "status": "no" }, { "lemma": "VAL_SUM_EQ", "status": "no" } ]
{ "lo": -5.14, "mid": 0.32, "hi": 6.51 }
4de797
modular_sum_quadratic_residues_v1_677425708_3407
Let $p$ be the largest prime number $n$ such that $2 \leq n \leq 354$. Compute $\frac{p(p-1)}{4}$.
31,064
graphs = [ Graph( let={ "_n": Const(2), "p": MaxOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Ref("_n")), Leq(Var("n"), Const(354)), IsPrime(Var("n"))))), "result": Div(Mul(Ref("p"), Sub(Ref("p"), Const(1))), Const(4)), }, goal=Ref("re...
NT
null
SUM
sympy
MAX_PRIME_BELOW
[ "MAX_PRIME_BELOW" ]
dc3ad3
modular_sum_quadratic_residues_v1
null
3
0
[ "MAX_PRIME_BELOW" ]
1
0.003
2026-02-08T05:41:24.748097Z
{ "verified": true, "answer": 31064, "timestamp": "2026-02-08T05:41:24.750661Z" }
dcd39b
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 81, "completion_tokens": 679 }, "timestamp": "2026-02-12T13:54:35.446Z", "answer": 31064 }, { ...
1
[ { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "K15", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, { "lemma": "MOD_MUL", "status": "...
{ "lo": -5.55, "mid": -3.01, "hi": 0.32 }
ca151c
modular_mod_compute_v1_1439011603_1857
Let $m$ be the minimum value of $x + y$ over all pairs of positive integers $(x, y)$ such that $xy = 1000000$. Find the remainder when $-39204$ is divided by $m$.
796
graphs = [ Graph( let={ "a": Const(-39204), "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")), Const(1000000)))), expr=Sum(Var("x"), Var("y")...
NT
null
COMPUTE
sympy
B3
[ "B3" ]
0cd20d
modular_mod_compute_v1
null
4
0
[ "B3" ]
1
0.002
2026-02-08T16:19:44.445541Z
{ "verified": true, "answer": 796, "timestamp": "2026-02-08T16:19:44.447982Z" }
c268b1
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 96, "completion_tokens": 734 }, "timestamp": "2026-02-17T01:38:14.211Z", "answer": 796 }, { ...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "V1", "status": "no" }, { "lemma": "V8_SUM", "status": ...
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
ee111b
sequence_fibonacci_compute_v1_1520064083_10018
Let $n$ be the smallest positive integer divisor of $640987$ that is at least as large as the number of positive integers $p$ for which there exists a positive integer $q$ such that $p \cdot q = 72$, $\gcd(p, q) = 1$, and $p < q$. Compute the $n$-th Fibonacci number.
28,657
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=72)), Eq(left=GCD(a=Var(name='p'), b=Var(name='q')), right=Const(value=1)), ...
NT
null
COMPUTE
sympy
COPRIME_PAIRS
[ "COPRIME_PAIRS/MIN_PRIME_FACTOR" ]
52cee2
sequence_fibonacci_compute_v1
null
6
0
[ "COPRIME_PAIRS", "MIN_PRIME_FACTOR" ]
2
0.002
2026-02-08T11:09:11.060435Z
{ "verified": true, "answer": 28657, "timestamp": "2026-02-08T11:09:11.061941Z" }
e964fe
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 123, "completion_tokens": 2395 }, "timestamp": "2026-02-14T10:35:20.958Z", "answer": 28657 }, ...
1
[ { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "DS2", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MIN_PRIME_FACTOR", "status": "ok_later" }, { "lemma": "V5", "status": "no" }, { "lemma": "V7", "status": "no" },...
{ "lo": -5.14, "mid": 0.32, "hi": 6.51 }
efa4bb
sequence_count_fib_divisible_v1_1439011603_1937
Let $\text{upper}$ be the number of integers $t$ with $19 \leq t \leq 951$ such that there exist integers $a$ and $b$ with $1 \leq a \leq 98$, $1 \leq b \leq 150$, and $t = 5a + 3b + 11$. Let $d = 4$. Compute the number of positive integers $n$ with $1 \leq n \leq \text{upper}$ such that the $n$-th Fibonacci number is ...
154
graphs = [ Graph( let={ "upper": CountOverSet(set=SolutionsSet(var=Var("t"), condition=Exists(var=Var(name='a'), condition=Exists(var=Var(name='b'), condition=And(Geq(left=Var(name='a'), right=Const(value=1)), Leq(left=Var(name='a'), right=Const(value=98)), Geq(left=Var(name='b'), right=Const(va...
NT
null
COUNT
sympy
LIN_FORM
[ "LIN_FORM" ]
7b2633
sequence_count_fib_divisible_v1
null
6
0
[ "LIN_FORM" ]
1
0.122
2026-02-08T16:23:23.157790Z
{ "verified": true, "answer": 154, "timestamp": "2026-02-08T16:23:23.279397Z" }
fca259
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 164, "completion_tokens": 3646 }, "timestamp": "2026-02-17T03:53:55.003Z", "answer": 154 }, { ...
1
[ { "lemma": "K18", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" }, { "lemma": "MOD_SUB", "status": "no" }, { "lemma": "V8_SUM", "status": "no" }, { "lemma": "VAL_SUM_EQ", "status": "no" } ...
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
1c6b3d
comb_count_surjections_v1_1520064083_3605
Let $k$ be the number of integers $t$ in the range $5 \leq t \leq 12$ for which there exist integers $a$ and $b$ such that $1 \leq a \leq 2$, $1 \leq b \leq 3$, and $t = 3a + 2b$. Let $n = 6$. Determine the value of $k! \cdot S(n, k)$, where $S(n, k)$ denotes the number of ways to partition a set of $n$ elements into $...
720
graphs = [ Graph( let={ "n": Const(6), "k": 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(nam...
COMB
null
COUNT
sympy
LIN_FORM
[ "LIN_FORM" ]
7b2633
comb_count_surjections_v1
null
4
0
[ "LIN_FORM" ]
1
0.003
2026-02-08T05:46:49.327783Z
{ "verified": true, "answer": 720, "timestamp": "2026-02-08T05:46:49.330362Z" }
dc0806
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 239, "completion_tokens": 1073 }, "timestamp": "2026-02-24T04:27:55.424Z", "answer": 720 }, { "id...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", "status": "no" }, { "lemma": "V8_SUM", "status": "no" } ]
{ "lo": -7.18, "mid": -5, "hi": -3.01 }
310cbe
modular_modexp_compute_v1_784195855_8490
Let $a = 43$ and let $e$ be the number of ordered pairs $(x, y)$ of integers such that $1 \le x \le 88$ and $1 \le y \le 88$. Let $m = 25200$. Compute the remainder when $a^e$ is divided by $m$.
16,801
graphs = [ Graph( let={ "a": Const(43), "e": CountOverSet(set=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(88)), right=IntegerRange(start=Const(1), end=Const(88)))), "m": Const(25200), "result": ModExp(base=Ref("a"), exp=Ref("e"), mod=Ref("m"))...
NT
null
COMPUTE
sympy
COUNT_CARTESIAN
[ "COUNT_CARTESIAN" ]
409d7d
modular_modexp_compute_v1
null
3
0
[ "COUNT_CARTESIAN" ]
1
0.002
2026-02-08T16:07:14.666818Z
{ "verified": true, "answer": 16801, "timestamp": "2026-02-08T16:07:14.668880Z" }
878418
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": 1739 }, "timestamp": "2026-02-16T21:06:59.122Z", "answer": 16801 }, ...
1
[ { "lemma": "COUNT_CARTESIAN", "status": "ok" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "K13", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "L3b", "status": "no" } ]
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
4ce29d
lin_form_endings_v1_1874849503_974
Let $t$ be an integer satisfying $70 \leq t \leq 1070$. Determine the number of such $t$ for which there exist integers $a$ and $b$ with $1 \leq a \leq 5$, $1 \leq b \leq 23$, and $t = 30a + 40b$. Multiply this number by 8191, and let the result be $S$. Find the remainder when $S$ is divided by 79502.
62,627
graphs = [ Graph( let={ "_inner_result": 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=5)), Geq(left=Var(name='b'), right=C...
COMB
null
COMPUTE
sympy
LIN_FORM
[ "LIN_FORM" ]
7b2633
lin_form_endings_v1
null
5
null
[ "LIN_FORM" ]
1
0.002
2026-02-08T13:29:41.079466Z
{ "verified": true, "answer": 62627, "timestamp": "2026-02-08T13:29:41.081245Z" }
b61203
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": 5760 }, "timestamp": "2026-02-24T18:34:32.115Z", "answer": 62627 }, { "...
1
[ { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "V8", "status": "no" }, { "lemma": "V8_SUM", ...
{ "lo": -2.35, "mid": 1.22, "hi": 4.84 }
19a7e7_n
alg_sym_quad_system_v1_1218484723_3621
An engineer is designing a triangular truss with side labels $a$, $b$, and $c$, all positive integers. The geometry of the truss requires $$a^{2} + b^{2} + c^{2} = ab + bc + ca.$$ The total load on the three joints must equal a specific count: let this count be the number of integers $n$ with $1 \le n \le 53982$ such ...
27,310
ALG
null
COMPUTE
sympy
C5
[ "C5", "B3" ]
2a47df
alg_sym_quad_system_v1
null
7
null
[ "B3", "C5" ]
2
0.018
2026-02-25T05:14:24.249795Z
null
83cbbe
19a7e7
narrative
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 381, "completion_tokens": 6477 }, "timestamp": "2026-03-30T20:20:54.229Z", "answer": 10 }, { "...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "C5", "status": "ok" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" } ]
{ "lo": 1.5, "mid": 4.69, "hi": 7.23 }
39476a
diophantine_product_count_v1_458359167_176
Let $n$ range over the positive integers from 1 to 5544. Define $\text{upper}$ to be the number of such $n$ for which the $n$th Fibonacci number is divisible by 24. Let $k = 720$. Now consider the set of positive integers $x$ such that $1 \leq x \leq \text{upper}$, $x$ divides $k$, and $\frac{k}{x} \leq \text{upper}$. ...
28
graphs = [ Graph( let={ "_n": Const(24), "k": Const(720), "upper": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), Const(5544)), Divides(divisor=Ref("_n"), dividend=Fibonacci(arg=Var(name='n')))))), "result": Co...
NT
null
COUNT
sympy
MOBIUS_COPRIME
[ "COUNT_FIB_DIVISIBLE" ]
66de3c
diophantine_product_count_v1
null
7
0
[ "COUNT_FIB_DIVISIBLE", "MOBIUS_COPRIME" ]
2
0.149
2026-02-08T03:03:09.481740Z
{ "verified": true, "answer": 28, "timestamp": "2026-02-08T03:03:09.631162Z" }
c4f15f
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 223, "completion_tokens": 6520 }, "timestamp": "2026-02-10T12:32:11.042Z", "answer": 28 }, { "id"...
1
[ { "lemma": "COUNT_FIB_DIVISIBLE", "status": "ok" }, { "lemma": "L3b", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MAX_VAL", "status": "no" }, { "lemma": "V3", "status": "no" } ...
{ "lo": -0.39, "mid": 1.8, "hi": 3.62 }
9c441f
diophantine_fbi2_count_v1_168721529_1749
Let $T$ be the set of all integers $t$ such that there exist positive integers $a$, $b$ with $1 \leq a \leq 15$, $1 \leq b \leq 8$, and $t = 6a + 15b$, and $21 \leq t \leq 210$. Let $N$ be the number of elements in $T$. Find the number of positive integers $d$ such that $6 \leq d \leq N$, $d$ divides $420$, and $2 \leq...
11
graphs = [ Graph( let={ "_n": Const(2), "k": Const(420), "result": CountOverSet(set=SolutionsSet(var=Var("d"), condition=And(Geq(Var("d"), Const(6)), Leq(Var("d"), CountOverSet(set=SolutionsSet(var=Var("t"), condition=Exists(var=Var(name='a'), condition=Exists(var=Var(nam...
NT
null
COUNT
sympy
C3
[ "LIN_FORM" ]
7b2633
diophantine_fbi2_count_v1
null
6
0
[ "C3", "LIN_FORM" ]
2
5.67
2026-02-08T13:54:15.067082Z
{ "verified": true, "answer": 11, "timestamp": "2026-02-08T13:54:20.736975Z" }
9abc1e
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 250, "completion_tokens": 7925 }, "timestamp": "2026-02-09T21:11:07.317Z", "answer": 11 }, { "id"...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "K13", "status": "no" }, { "lemma": "L3b", "status": "no" }, { "lemma": "L3c", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "V8", "status": "no" } ]
{ "lo": -5.49, "mid": 0.03, "hi": 6.12 }
e1a6e0
antilemma_cartesian_v1_1915831931_254
Let $x$ be the number of ordered pairs $(i, j)$ such that $1 \leq i \leq 11$ and $1 \leq j \leq 13$. Let $c = 625$. Compute $c - x$.
482
graphs = [ Graph( let={ "x": CountOverSet(set=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(11)), right=IntegerRange(start=Const(1), end=Const(13)))), "_c": Const(625), "Q": Sub(Ref("_c"), Ref("x")), }, goal=Ref("Q"), ) ]
COMB
GEOM
COMPUTE
sympy
COUNT_CARTESIAN
[ "COUNT_CARTESIAN" ]
409d7d
antilemma_cartesian_v1
null
2
0
[ "COUNT_CARTESIAN" ]
1
0.001
2026-02-08T15:17:13.143497Z
{ "verified": true, "answer": 482, "timestamp": "2026-02-08T15:17:13.144317Z" }
b901ee
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 165, "completion_tokens": 272 }, "timestamp": "2026-02-24T20:32:07.972Z", "answer": 482 }, { "id"...
2
[ { "lemma": "COUNT_CARTESIAN", "status": "ok" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "V8", "status": "no" }, { "lemma": "V8_SU...
{ "lo": -10, "mid": -7.42, "hi": -4.85 }
65d106
nt_min_coprime_above_v1_717093673_2191
Let $a$ be the smallest integer $n$ such that $86436 < n \leq 86768$ and $\gcd(n, 322) = 1$. Let $b$ be the minimum value of $x + y$ over all ordered pairs of positive integers $(x, y)$ such that $xy = 25\,000\,000$. Compute the remainder when $b - a$ is divided by $86654$.
10,217
graphs = [ Graph( let={ "start": Const(86436), "upper": Const(86768), "modulus": Const(322), "result": MinOverSet(set=SolutionsSet(var=Var("n"), condition=And(Gt(Var("n"), Ref("start")), Leq(Var("n"), Ref("upper")), Eq(GCD(a=Var("n"), b=Ref("modulus")), Const(...
NT
null
EXTREMUM
sympy
B3
[ "B3" ]
fc629c
nt_min_coprime_above_v1
negation_mod
3
0
[ "B3" ]
1
0.054
2026-02-08T16:36:36.178707Z
{ "verified": true, "answer": 10217, "timestamp": "2026-02-08T16:36:36.232437Z" }
0ea88c
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": 2106 }, "timestamp": "2026-02-17T08:19:41.970Z", "answer": 10217 }, ...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" }, { "lemma": "MAX_VAL", "status": "no" }, { "lemma": "MOD_MUL", "status"...
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
219618
comb_factorial_compute_v1_151522320_1413
Let $m = 23$. Define $k$ to be the number of integers $n$ with $1 \leq n \leq m$ such that $n \equiv \left\lfloor \frac{n}{2} \right\rfloor \pmod{3}$. Let $S$ be the set of all positive integers $p$ for which there exists a positive integer $q > p$ such that $pq = 12$ and $\gcd(p, q) = 1$. Define $n$ to be the largest ...
5,040
graphs = [ Graph( let={ "_m": Const(23), "_n": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), Ref("_m")), Congruent(a=Var(name='n'), b=Floor(arg=Div(left=Var(name='n'), right=Const(value=2))), modulus=Const(value=3))))), "...
NT
null
COMPUTE
sympy
COPRIME_PAIRS
[ "COPRIME_PAIRS/MAX_PRIME_BELOW", "L3C/MAX_PRIME_BELOW" ]
05d7f9
comb_factorial_compute_v1
null
6
0
[ "COPRIME_PAIRS", "L3C", "MAX_PRIME_BELOW" ]
3
0.003
2026-02-08T03:59:38.913299Z
{ "verified": true, "answer": 5040, "timestamp": "2026-02-08T03:59:38.915930Z" }
0bbfa3
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 247, "completion_tokens": 1552 }, "timestamp": "2026-02-11T16:12:07.788Z", "answer": 5040 }, { "i...
1
[ { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "K18", "status": "no" }, { "lemma": "L3C", "status": "ok" }, { "lemma": "L3c", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok_later"...
{ "lo": -5.55, "mid": -3.02, "hi": 0.31 }
d98caa
modular_min_linear_v1_1918700295_2506
Let $a = \sum_{d \mid 2649} \phi(d)$, where $\phi$ is Euler's totient function. Let $x$ be the smallest positive integer such that $x \leq 3394$ and $a \cdot x \equiv 92 \pmod{3394}$. Compute the remainder when $44121 \cdot x$ is divided by $70505$.
38,558
graphs = [ Graph( let={ "_n": Const(2649), "a": SumOverDivisors(n=Ref(name='_n'), var='d', expr=EulerPhi(n=Var(name='d'))), "b": Const(92), "m": Const(3394), "result": MinOverSet(set=SolutionsSet(var=Var("x"), condition=And(Geq(Var("x"), Const(1)),...
NT
null
EXTREMUM
sympy
K3
[ "K3" ]
54c41e
modular_min_linear_v1
null
6
0
[ "K3" ]
1
0.252
2026-02-08T07:56:11.331864Z
{ "verified": true, "answer": 38558, "timestamp": "2026-02-08T07:56:11.583896Z" }
9b60f0
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 129, "completion_tokens": 3102 }, "timestamp": "2026-02-13T13:49:19.615Z", "answer": 38558 }, ...
1
[ { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "K14", "status": "no" }, { "lemma": "K3", "status": "ok" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", "status": "no" }, { "lemma": "VAL_SUM_EQ", "status": "no" } ]
{ "lo": -3.44, "mid": 1.34, "hi": 6.56 }
e96ce5
nt_count_primes_v1_784195855_2339
Compute the number of ordered pairs $(p, q)$ of positive integers such that $pq = 18$, $\gcd(p, q) = 1$, and $p < q$. Let this count be $L$. Compute the number of prime numbers $n$ such that $L \leq n \leq 45369$.
4,707
graphs = [ Graph( let={ "upper": Const(45369), "result": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), CountOverSet(set=SolutionsSet(var=Var("p"), condition=And(IsPositive(arg=Var(name='p')), Exists(var=Var(name='q'), condition=And(Eq(left=Mul(Var(name='p'),...
NT
null
COUNT
sympy
COPRIME_PAIRS
[ "COPRIME_PAIRS" ]
2bb3aa
nt_count_primes_v1
null
4
0
[ "COPRIME_PAIRS" ]
1
9.058
2026-02-08T05:41:41.505658Z
{ "verified": true, "answer": 4707, "timestamp": "2026-02-08T05:41:50.564019Z" }
9c8471
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 114, "completion_tokens": 2024 }, "timestamp": "2026-02-12T13:08:07.603Z", "answer": 4707 }, {...
1
[ { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" }, { "lemma": "V5", "status": "no" }, { "lemma": "V8_SUM", "stat...
{ "lo": -3.44, "mid": 1.34, "hi": 6.56 }
6d9e20
diophantine_product_count_v1_50713871_85
Let $k$ be the number of integers $t$ such that $7 \leq t \leq 430$ and there exist positive integers $a \leq 58$ and $b \leq 70$ satisfying $t = 5a + 2b$. Let $u = 332$. Define $r$ to be the number of positive integers $x \leq u$ such that $x$ divides $k$ and $\frac{k}{x} \leq u$. Find the value of $65361 \cdot r \bmo...
33,406
graphs = [ Graph( let={ "_n": Const(50162), "k": 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=58)), Geq(left=V...
NT
null
COUNT
sympy
LIN_FORM
[ "LIN_FORM" ]
7b2633
diophantine_product_count_v1
null
6
0
[ "LIN_FORM" ]
1
0.019
2026-02-08T02:44:50.688752Z
{ "verified": true, "answer": 33406, "timestamp": "2026-02-08T02:44:50.708220Z" }
060c75
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 238, "completion_tokens": 6140 }, "timestamp": "2026-02-08T19:48:27.980Z", "answer": 32606 }, { ...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "K15", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "MOD_ADD", "status": "no" }, { "lemma": "MOD_SUB", "status": "no" }, { "lemma": "VAL_SUM_EQ", "status": "no" } ]
{ "lo": 1.96, "mid": 3.89, "hi": 5.66 }
a4c9a2
comb_factorial_compute_v1_601307018_8827
Let $n = \sum_{k=0}^{2} 2^k$ and let $M = n!$. Find the remainder when $44121M$ is divided by $99538$.
1,948
graphs = [ Graph( let={ "_n": Const(99538), "n": Summation(var="k", start=Const(0), end=Const(2), expr=Pow(Const(2), Var("k"))), "result": Factorial(Ref("n")), "Q": Mod(value=Mul(Const(44121), Ref("result")), modulus=Ref("_n")), }, goal=Ref("Q"...
COMB
null
COMPUTE
sympy
SUM_GEOM
[ "SUM_GEOM" ]
04214c
comb_factorial_compute_v1
null
2
0
[ "SUM_GEOM" ]
1
0.002
2026-03-10T09:16:56.643258Z
{ "verified": true, "answer": 1948, "timestamp": "2026-03-10T09:16:56.645430Z" }
44a29b
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 152, "completion_tokens": 1059 }, "timestamp": "2026-04-19T09:54:23.469Z", "answer": 1948 }, { "...
1
[ { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "SUM_GEOM", "status": "ok" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8_SUM", ...
{ "lo": -5.34, "mid": 1.1, "hi": 7.53 }
342d9a
nt_sum_divisors_mod_v1_124444284_9837
Let $S$ be the set of all ordered pairs $(x, y)$ of positive integers such that $xy = 6350400$. For each pair $(x, y)$, compute $x + y$. Let $n$ be the minimum value of $x + y$ over all such pairs. Let $\sigma$ denote the sum of all positive divisors of $n$. Find the remainder when $\sigma$ is divided by $10247$.
9,097
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")), Const(6350400)))), expr=Sum(Var("x"), Var("y")))), "M": Const(1024...
NT
null
COMPUTE
sympy
B3
[ "B3" ]
0cd20d
nt_sum_divisors_mod_v1
null
6
0
[ "B3" ]
1
0.002
2026-02-08T12:42:02.516444Z
{ "verified": true, "answer": 9097, "timestamp": "2026-02-08T12:42:02.518603Z" }
ef395a
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": 5070 }, "timestamp": "2026-02-15T04:09:17.661Z", "answer": 9097 }, {...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "K15", "status": "no" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", "status": "no" } ]
{ "lo": -5.14, "mid": 0.32, "hi": 6.51 }
692ef9
antilemma_k3_v1_784195855_7265
Let $n = 25615$. Define $x = \sum_{d \mid n} \phi(d)$, where $\phi(d)$ denotes Euler's totient function. Compute $x$.
25,615
graphs = [ Graph( let={ "_n": Const(25615), "x": SumOverDivisors(n=Ref(name='_n'), var='d', expr=EulerPhi(n=Var(name='d'))), }, goal=Ref("x"), ) ]
NT
COMB
COMPUTE
sympy
K3
[ "K3" ]
54c41e
antilemma_k3_v1
null
2
0
[ "K3" ]
1
0
2026-02-08T09:10:17.188299Z
{ "verified": true, "answer": 25615, "timestamp": "2026-02-08T09:10:17.188655Z" }
8b1c23
CC BY 4.0
[ { "id": 8, "model": "mathstral", "score": 1, "correct": { "strict": false, "boxed": false, "relaxed": true }, "usage": { "prompt_tokens": 102, "completion_tokens": 617 }, "timestamp": "2026-02-15T20:35:19.631Z", "answer": 10240 }, { "id": 11,...
1
[ { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "K3", "status": "ok" }, { "lemma": "K5", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" }, { "lemma": "V7", "status": "no" } ]
{ "lo": -5.55, "mid": -3.01, "hi": 0.34 }
40ad33
nt_sum_totient_over_divisors_v1_865884756_1614
Let $n$ be the sum of $\phi(d)$ over all positive divisors $d$ of $2520$, where $\phi$ denotes Euler's totient function. Let $S$ be the sum of $\phi(d_1)$ over all positive divisors $d_1$ of $n$. Compute $S$.
2,520
graphs = [ Graph( let={ "_n": Const(2520), "n": SumOverDivisors(n=Ref(name='_n'), var='d', expr=EulerPhi(n=Var(name='d'))), "result": SumOverDivisors(n=Ref(name='n'), var='d1', expr=EulerPhi(n=Var(name='d1'))), }, goal=Ref("result"), ) ]
NT
null
COMPUTE
sympy
K3
[ "K3" ]
54c41e
nt_sum_totient_over_divisors_v1
null
3
0
[ "K3" ]
1
0.012
2026-02-08T16:11:26.136039Z
{ "verified": true, "answer": 2520, "timestamp": "2026-02-08T16:11:26.148500Z" }
1ee60a
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 112, "completion_tokens": 323 }, "timestamp": "2026-02-16T23:02:22.397Z", "answer": 2520 }, { ...
1
[ { "lemma": "K3", "status": "ok" }, { "lemma": "K5", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" } ]
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
49ce84
diophantine_fbi2_count_v1_1978505735_524
Let $n = 54$. Let $k$ be the number of integers $t$ such that $9 \leq t \leq 74$ and there exist integers $a$ and $b$ with $1 \leq a \leq 6$, $1 \leq b \leq 16$, and $t = 7a + 2b$. Let $r$ be the number of divisors $d$ of $k$ such that $5 \leq d \leq 54$ and $5 \leq \frac{k}{d} \leq 54$. Let $c$ be the largest prime nu...
13,388
graphs = [ Graph( let={ "_n": Const(54), "k": CountOverSet(set=SolutionsSet(var=Var("t"), condition=Exists(var=Var(name='a'), condition=Exists(var=Var(name='b'), condition=And(Geq(left=Var(name='a'), right=Const(value=1)), Leq(left=Var(name='a'), right=Const(value=6)), Geq(left=Var(n...
NT
null
COUNT
sympy
MAX_PRIME_BELOW
[ "MAX_PRIME_BELOW", "LIN_FORM" ]
4a5bfb
diophantine_fbi2_count_v1
affine_mod
5
0
[ "LIN_FORM", "MAX_PRIME_BELOW" ]
2
0.015
2026-02-08T15:26:05.836879Z
{ "verified": true, "answer": 13388, "timestamp": "2026-02-08T15:26:05.851488Z" }
cfd14a
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 191, "completion_tokens": 2926 }, "timestamp": "2026-02-16T05:46:04.198Z", "answer": 13388 }, ...
1
[ { "lemma": "K16", "status": "no" }, { "lemma": "L3b", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, { "lemma": "V1", "status": "no" }, { ...
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
176866
comb_count_derangements_v1_677425708_1480
Let $n$ be the number of nonnegative integers $j$ such that $0 \leq j \leq 45056$ and $\binom{45056}{j}$ is odd. Compute the subfactorial of $n$, denoted $!n$, which is the number of derangements of $n$ elements.
14,833
graphs = [ Graph( let={ "_n": Const(45056), "n": CountOverSet(set=SolutionsSet(var=Var("j"), condition=And(Geq(Var("j"), Const(0)), Leq(Var("j"), Const(45056)), Eq(Mod(value=Binom(n=Ref("_n"), k=Var("j")), modulus=Const(2)), Const(1))), domain='nonnegative_integers')), "r...
COMB
null
COUNT
sympy
V8
[ "V8" ]
86348e
comb_count_derangements_v1
null
6
0
[ "V8" ]
1
0.001
2026-02-08T04:13:36.412853Z
{ "verified": true, "answer": 14833, "timestamp": "2026-02-08T04:13:36.413944Z" }
0255ac
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 180, "completion_tokens": 1226 }, "timestamp": "2026-02-09T20:32:59.144Z", "answer": 14833 }, { "...
1
[ { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "K14", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "V8", "status...
{ "lo": -2.46, "mid": 0.47, "hi": 3.59 }
bdb38f_n
sequence_fibonacci_compute_v1_601307018_862
A composer assigns musical motifs based on Fibonacci and Bell numbers. The 24th Fibonacci number determines a seed value $M$. The number of ways to write 22 as a sum of two positive odd integers gives a modulus $k$. The final motif index is the Bell number $B_{M \bmod k}$. Compute this index.
5
ALG
COMB
COMPUTE
sympy
STARS_BARS
[ "COMB1" ]
d93ba8
sequence_fibonacci_compute_v1
bell_mod
5
null
[ "COMB1", "STARS_BARS" ]
2
0.508
2026-03-10T01:28:33.954188Z
null
b18287
bdb38f
narrative
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 1, "correct": { "strict": false, "boxed": false, "relaxed": true }, "usage": { "prompt_tokens": 179, "completion_tokens": 1804 }, "timestamp": "2026-03-29T14:40:11.366Z", "answer": 1 }, { "id...
1
[ { "lemma": "COMB1", "status": "ok" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" } ]
{ "lo": 1.7, "mid": 5.03, "hi": 8.38 }
9d7b9d
modular_min_linear_v1_1874849503_137
Let $S$ be the set of all integers $n$ such that $1 \le n \le p$, where $p$ is the largest prime number less than or equal to $5853$. Define $a$ to be the number of integers in $S$ that are relatively prime to $30$. Let $m = 18643$ and $b = 1261$. Determine the value of the smallest positive integer $x$ such that $1 \l...
15,252
graphs = [ Graph( let={ "a": 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"), Const(2)), Leq(Var("n"), Const(5853)), IsPrime(Var("n")))))), Eq(GCD(a=Var("n"), b=Const(30)), Const...
NT
null
EXTREMUM
sympy
MAX_PRIME_BELOW
[ "MAX_PRIME_BELOW/C4" ]
a99ef8
modular_min_linear_v1
null
6
0
[ "C4", "MAX_PRIME_BELOW" ]
2
1.891
2026-02-08T12:49:45.525631Z
{ "verified": true, "answer": 15252, "timestamp": "2026-02-08T12:49:47.416859Z" }
ac58de
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 226, "completion_tokens": 5821 }, "timestamp": "2026-02-10T03:33:20.774Z", "answer": 15252 }, { ...
1
[ { "lemma": "C4", "status": "ok_later" }, { "lemma": "K13", "status": "no" }, { "lemma": "K15", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "K18", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "MAX_PRIM...
{ "lo": -5.13, "mid": 0.83, "hi": 7.52 }
e14879
geo_count_lattice_rect_v1_1978505735_2103
Compute the number of lattice points $(x, y)$ such that $0 \leq x \leq 180$ and $0 \leq y \leq 221$.
40,182
graphs = [ Graph( let={ "a": Const(180), "b": Const(221), "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
0.001
2026-02-08T16:39:41.590196Z
{ "verified": true, "answer": 40182, "timestamp": "2026-02-08T16:39:41.590826Z" }
a8f350
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 81, "completion_tokens": 1084 }, "timestamp": "2026-02-24T21:46:10.428Z", "answer": 40182 }, {...
1
[]
{ "lo": -6.4, "mid": -4.13, "hi": -2.01 }
4d1109
nt_gcd_compute_v1_1978505735_300
Let $a = 118641$ and $b = 276829$. Let $d$ be the greatest common divisor of $a$ and $b$. Compute $d$.
39,547
graphs = [ Graph( let={ "a": Const(118641), "b": Const(276829), "result": GCD(a=Ref("a"), b=Ref("b")), }, goal=Ref("result"), ) ]
NT
null
COMPUTE
sympy
LIN_FORM
[ "LIN_FORM/C4/MOBIUS_SQUAREFREE", "DIVISOR_PARITY" ]
46bdbd
nt_gcd_compute_v1
null
2
0
[ "C4", "DIVISOR_PARITY", "LIN_FORM", "MOBIUS_SQUAREFREE" ]
4
0.007
2026-02-08T15:17:41.120420Z
{ "verified": true, "answer": 39547, "timestamp": "2026-02-08T15:17:41.127097Z" }
13743c
CC BY 4.0
[ { "id": 8, "model": "mathstral", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 102, "completion_tokens": 419 }, "timestamp": "2026-02-16T05:21:38.901Z", "answer": 39547 }, { "id": 11, ...
2
[ { "lemma": "C4", "status": "ok_later" }, { "lemma": "DIVISOR_PARITY", "status": "ok" }, { "lemma": "K14", "status": "no" }, { "lemma": "K18", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" ...
{ "lo": -10, "mid": -7.27, "hi": -4.54 }
9edca0
nt_lcm_compute_v1_124444284_6784
Let $a = 1095$ and let $b$ be the number of positive integers $n \leq 8284$ such that $4$ divides $n$ and $\gcd(n, 15) = 1$. Let $\text{result} = \text{lcm}(a, b)$. Let $S$ be the set of all ordered pairs $(x, y)$ of positive integers such that $xy = 64$, and let $m = \min\{x + y \mid (x, y) \in S\}$. Compute the remai...
73,127
graphs = [ Graph( let={ "_n": Const(8284), "a": Const(1095), "b": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), Ref("_n")), Divides(divisor=Const(4), dividend=Var("n")), Eq(GCD(a=Var("n"), b=Const(15)), Const(1))))), ...
NT
null
COMPUTE
sympy
B3
[ "B3", "C5" ]
cc81ff
nt_lcm_compute_v1
mod_exp
6
0
[ "B3", "C5" ]
2
0.002
2026-02-08T08:38:17.434754Z
{ "verified": true, "answer": 73127, "timestamp": "2026-02-08T08:38:17.437189Z" }
c384c7
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 190, "completion_tokens": 1423 }, "timestamp": "2026-02-13T20:12:24.153Z", "answer": 73127 }, ...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "C5", "status": "ok" }, { "lemma": "K16", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "V3", "status": "no" }, { "lemma": "V5", "status": "no" }, { "lemma": "V8", "status"...
{ "lo": -3.51, "mid": 1.33, "hi": 6.56 }
601e8d_n
geo_visible_lattice_v1_1419126231_943
A city planner designs a $196 \times 196$ grid of city blocks, each at integer coordinates $(x,y)$ with $1 \leq x, y \leq 196$. A diagonal walkway connects the southwest corner of block $(1,1)$ to the northeast corner of block $(x,y)$. The walkway is considered *unobstructed* if no other grid point lies on the segment ...
54,527
GEOM
GEOM
COUNT
sympy
[]
geo_visible_lattice_v1
null
3
null
null
null
0.769
2026-02-25T10:27:09.365336Z
null
5c5b28
601e8d
narrative
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 253, "completion_tokens": 13923 }, "timestamp": "2026-03-31T04:09:25.980Z", "answer": 54103 }, { ...
1
[]
{ "lo": 1.5, "mid": 4.69, "hi": 7.23 }
5bfee7
comb_bell_compute_v1_865884756_1262
Let $n$ be the number of positive integers $p$ for which there exists a positive integer $q$ such that $pq = 220500$, $\gcd(p, q) = 1$, and $p < q$. Let $B_n$ denote the $n$th Bell number, which is the number of partitions of a set of $n$ elements. Compute the remainder when $69163 \cdot B_n$ is divided by $78387$.
65,496
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=220500)), Eq(left=GCD(a=Var(name='p'), b=Var(name='q')), right=Const(value=1)...
NT
COMB
COMPUTE
sympy
COPRIME_PAIRS
[ "COPRIME_PAIRS" ]
2bb3aa
comb_bell_compute_v1
null
6
0
[ "COPRIME_PAIRS" ]
1
0.002
2026-02-08T15:51:36.148417Z
{ "verified": true, "answer": 65496, "timestamp": "2026-02-08T15:51:36.150167Z" }
91f8b3
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 143, "completion_tokens": 1814 }, "timestamp": "2026-02-16T14:23:16.705Z", "answer": 65496 }, ...
1
[ { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "K13", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "V1", "status": "no" }, { "lemma": "V3", "status": "no" } ]
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
ab52e1
modular_count_residue_v1_784195855_5026
Let $m$ be the number of positive integers $n$ such that $1 \le n \le 54$ and the sum of the digits of $n$ is even. Let $r$ be the smallest divisor of $41327$ that is at least $2$. Compute the number of positive integers $n$ such that $1 \le n \le 51076$ and $n \equiv r \pmod{m}$.
1,892
graphs = [ Graph( let={ "_n": Const(2), "upper": Const(51076), "m": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), Const(54)), Eq(Mod(value=DigitSum(Var("n")), modulus=Const(2)), Const(0))))), "r": MinOverSet(s...
NT
null
COUNT
sympy
MIN_PRIME_FACTOR
[ "MIN_PRIME_FACTOR", "L3B" ]
50af68
modular_count_residue_v1
null
5
0
[ "L3B", "MIN_PRIME_FACTOR" ]
2
1.679
2026-02-08T07:35:48.485830Z
{ "verified": true, "answer": 1892, "timestamp": "2026-02-08T07:35:50.165284Z" }
c4615a
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 138, "completion_tokens": 1775 }, "timestamp": "2026-02-13T11:19:04.962Z", "answer": 1965 }, ...
1
[ { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "L3B", "status": "ok" }, { "lemma": "L3c", "status": "no" }, { "lemma": "MIN_PRIME_FACTOR", "statu...
{ "lo": -3.44, "mid": 1.34, "hi": 6.56 }
a0123b
nt_count_intersection_v1_458359167_4635
Let $N = 10000$. Let $b$ be the number of integers $t$ such that $7 \leq t \leq 30$ and there exist positive integers $a$ and $b$ with $1 \leq a \leq 4$, $1 \leq b \leq 5$, and $t = 5a + 2b$. Compute the number of positive integers $n \leq N$ such that $3$ divides $n$ and $\gcd(n, b) = 1$.
1,334
graphs = [ Graph( let={ "N": Const(10000), "a": Const(3), "b": 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=Co...
NT
null
COUNT
sympy
LIN_FORM
[ "LIN_FORM" ]
7b2633
nt_count_intersection_v1
null
5
0
[ "LIN_FORM" ]
1
0.472
2026-02-08T11:57:10.163809Z
{ "verified": true, "answer": 1334, "timestamp": "2026-02-08T11:57:10.635404Z" }
4b91f8
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 156, "completion_tokens": 2509 }, "timestamp": "2026-02-14T21:18:29.879Z", "answer": 1334 }, {...
1
[ { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "K13", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "V5", "status": "no" } ]
{ "lo": -6.96, "mid": -4.56, "hi": -1.46 }
5a6e16
diophantine_product_count_v1_1742523217_3317
Let $k$ be the sum of $\phi(d)$ over all positive divisors $d$ of $180$, where $\phi$ denotes Euler's totient function. Determine the number of positive integers $x$ such that $1 \leq x \leq 120$, $x$ divides $k$, and $\frac{k}{x} \leq 120$.
16
graphs = [ Graph( let={ "k": SumOverDivisors(n=Const(value=180), var='d', expr=EulerPhi(n=Var(name='d'))), "upper": Const(120), "result": CountOverSet(set=SolutionsSet(var=Var("x"), condition=And(Geq(Var("x"), Const(1)), Leq(Var("x"), Ref("upper")), Divides(divisor=Var("x...
NT
null
COUNT
sympy
K3
[ "K3" ]
54c41e
diophantine_product_count_v1
null
5
0
[ "K3" ]
1
0.012
2026-02-08T05:46:40.579970Z
{ "verified": true, "answer": 16, "timestamp": "2026-02-08T05:46:40.592141Z" }
187e68
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 120, "completion_tokens": 871 }, "timestamp": "2026-02-12T14:50:04.680Z", "answer": 16 }, { ...
1
[ { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "K13", "status": "no" }, { "lemma": "K3", "status": "ok" }, { "lemma": "LTE_DIFF_P2", "s...
{ "lo": -3.44, "mid": 1.34, "hi": 6.56 }
49141b
antilemma_count_primes_v1_1918700295_52
Compute the number of prime numbers $n$ such that $2 \leq n \leq 2887$.
418
graphs = [ Graph( let={ "x": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(2)), Leq(Var("n"), Const(2887)), IsPrime(Var("n"))))), }, goal=Ref("x"), ) ]
NT
null
COMPUTE
sympy
COUNT_PRIMES
[ "COUNT_PRIMES" ]
07c874
antilemma_count_primes_v1
null
6
0
[ "COUNT_PRIMES" ]
1
0
2026-02-08T02:57:54.738793Z
{ "verified": true, "answer": 418, "timestamp": "2026-02-08T02:57:54.739250Z" }
f8790b
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 138, "completion_tokens": 7714 }, "timestamp": "2026-02-08T22:20:03.013Z", "answer": 418 }, { "i...
1
[ { "lemma": "COUNT_PRIMES", "status": "ok" }, { "lemma": "DS2", "status": "no" }, { "lemma": "L3b", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" }, { "lemma": "V8_SUM", "status": "no" } ]
{ "lo": -6.49, "mid": 0.51, "hi": 7.52 }
e8cc85
antilemma_v7_kummer_798873815_167
Compute the largest integer $k$ such that $5^k$ divides $\binom{185}{74}$.
3
graphs = [ Graph( let={ "x": MaxKDivides(target=Binom(n=Const(185), k=Const(74)), base=Const(5)), }, goal=Ref("x"), ) ]
NT
null
COMPUTE
sympy
V7
[ "V7" ]
0672d4
antilemma_v7_kummer
null
6
null
[ "V7" ]
1
0.038
2026-02-08T02:29:55.429804Z
{ "verified": true, "answer": 3, "timestamp": "2026-02-08T02:29:55.467997Z" }
202bfb
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 138, "completion_tokens": 835 }, "timestamp": "2026-02-08T19:06:43.285Z", "answer": 3 }, { "id": ...
1
[ { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "K13", "status": "no" }, { "lemma": "V1", "status": "no" }, { "lemma": "V7", "status": "ok" }, { "lemma": "V8", "status": "no" } ]
{ "lo": -7.46, "mid": -4.17, "hi": -1.14 }
99e7e0
nt_count_divisible_and_v1_397696148_1924
Let $S$ be the set of all ordered pairs $(x, y)$ of positive integers such that $xy = 15681600$. Let $T$ be the set of all values $x + y$ for $(x, y) \in S$. Let $m$ be the minimum value in $T$. Compute the number of positive integers $n$ such that $1 \leq n \leq m$, $n$ is divisible by $6$, and $n$ is divisible by $10...
1,520
graphs = [ Graph( let={ "upper": MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var("x"), Var("y")]), condition=And(IsPositive(arg=Var(name='x')), IsPositive(arg=Var(name='y')), Eq(Mul(Var("x"), Var("y")), Const(15681600)))), expr=Sum(Var("x"), Var("y")))), "d1": Cons...
NT
null
COUNT
sympy
B3
[ "B3" ]
0cd20d
nt_count_divisible_and_v1
null
7
0
[ "B3" ]
1
0.545
2026-02-08T12:50:55.153089Z
{ "verified": true, "answer": 1520, "timestamp": "2026-02-08T12:50:55.697897Z" }
984b6d
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 191, "completion_tokens": 3332 }, "timestamp": "2026-02-15T06:40:08.971Z", "answer": 1520 }, {...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "MAX_VAL", "status": "no" }, { "lemma": "MOD_FACTORIAL", "status"...
{ "lo": -5.49, "mid": 0.22, "hi": 6.51 }
69167f
comb_factorial_compute_v1_601307018_361
Let $n$ be the number of ordered pairs $(a, b)$ of positive integers with $1 \le a, b \le 25$ satisfying $$ 64a^3 + 144a^2b + 27b^3 + C \cdot a b^2 = 658503, $$ where $C = \left|\left\{ t : t = 2a_1 + 7b_1\ \text{for some integers}\ a_1,b_1\ \text{with}\ 1 \le a_1 \le 412,\ 1 \le b_1 \le 85,\ \text{and}\ 9 \le t \le 14...
64,260
graphs = [ Graph( let={ "_m": Const(15), "_n": Const(67585), "n": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var("a"), Var("b")]), condition=And(Geq(Var("a"), Const(1)), Leq(Var("a"), Const(25)), Geq(Var("b"), Const(1)), Leq(Var("b"), Const(25)), Eq(Sum(Mul(Const(1...
COMB
null
COMPUTE
sympy
LIN_FORM
[ "LIN_FORM/QF_PSD_COUNT_LEQ/POLY3_COUNT" ]
fb45da
comb_factorial_compute_v1
null
7
0
[ "LIN_FORM", "POLY3_COUNT", "QF_PSD_COUNT_LEQ" ]
3
0.043
2026-03-10T00:54:19.674740Z
{ "verified": true, "answer": 64260, "timestamp": "2026-03-10T00:54:19.718121Z" }
2c9b8e
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 293, "completion_tokens": 32768 }, "timestamp": "2026-03-28T22:54:04.753Z", "answer": null }, { ...
0
[ { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "POLY3_COUNT", "status": "ok_later" }, { "lemma"...
{ "lo": 5.22, "mid": 7.83, "hi": 10 }
ce96a0
sequence_lucas_compute_v1_458359167_3288
Let $n$ be the smallest divisor of $437$ that is greater than or equal to $2$. Compute the $n$-th Lucas number.
9,349
graphs = [ Graph( let={ "_n": Const(437), "n": MinOverSet(set=SolutionsSet(var=Var("d"), condition=And(Geq(Var("d"), Const(2)), Divides(divisor=Var("d"), dividend=Ref("_n"))))), "result": Lucas(arg=Ref(name='n')), }, goal=Ref("result"), ) ]
NT
null
COMPUTE
sympy
MIN_PRIME_FACTOR
[ "MIN_PRIME_FACTOR" ]
bc3776
sequence_lucas_compute_v1
null
2
0
[ "MIN_PRIME_FACTOR" ]
1
0.001
2026-02-08T08:15:33.796169Z
{ "verified": true, "answer": 9349, "timestamp": "2026-02-08T08:15:33.797135Z" }
b98b43
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 78, "completion_tokens": 726 }, "timestamp": "2026-02-13T16:25:13.074Z", "answer": 9349 }, { ...
1
[ { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "K14", "status": "no" }, { "lemma": "K18", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" }, { "lemma": "MIN_PRIME_FACTOR", "status": "...
{ "lo": -7.01, "mid": -4.68, "hi": -2.15 }
8bfc74
antilemma_cartesian_v1_151522320_876
Let $x$ be the number of ordered pairs $(a,b)$ such that $1 \leq a \leq 16$ and $1 \leq b \leq 23$. Compute the smallest positive integer $k$ such that the $k$-th Fibonacci number is divisible by $x + 2$.
285
graphs = [ Graph( let={ "x": CountOverSet(set=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(16)), right=IntegerRange(start=Const(1), end=Const(23)))), "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
4
0
[ "COUNT_CARTESIAN" ]
1
0.001
2026-02-08T03:37:34.137479Z
{ "verified": true, "answer": 285, "timestamp": "2026-02-08T03:37:34.138395Z" }
689dc3
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 178, "completion_tokens": 2032 }, "timestamp": "2026-02-10T15:12:27.064Z", "answer": 285 }, { "id...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_CARTESIAN", "status": "ok" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "V8", ...
{ "lo": -2.46, "mid": 0.47, "hi": 3.59 }
6648ac
geo_visible_lattice_v1_1125832087_120
A lattice point $(x, y)$ is called visible from the origin if $\gcd(x, y) = 1$. Let $n = 77$. Compute the number of visible lattice points $(x, y)$ such that $1 \leq x \leq n$ and $1 \leq y \leq n$. Find the value of this number.
3,663
graphs = [ Graph( let={ "n": Const(77), "result": VisibleLatticePoints(n=Ref(name='n')), }, goal=Ref("result"), ) ]
GEOM
null
COUNT
sympy
[]
geo_visible_lattice_v1
null
6
0
null
null
0.127
2026-02-08T02:52:25.243270Z
{ "verified": true, "answer": 3663, "timestamp": "2026-02-08T02:52:25.369812Z" }
07c104
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": 2461 }, "timestamp": "2026-02-10T11:46:53.025Z", "answer": 3663 }, { "i...
1
[]
{ "lo": 2.57, "mid": 4, "hi": 5.3 }
a3b7bb
antilemma_k3_v1_1978505735_1084
Compute the sum of $\varphi(d)$ over all positive divisors $d$ of $13108$, where $\varphi$ denotes Euler's totient function.
13,108
graphs = [ Graph( let={ "_n": Const(13108), "x": SumOverDivisors(n=Ref(name='_n'), var='d', expr=EulerPhi(n=Var(name='d'))), }, goal=Ref("x"), ) ]
NT
COMB
COMPUTE
sympy
K3
[ "K3" ]
54c41e
antilemma_k3_v1
null
2
0
[ "K3" ]
1
0.001
2026-02-08T15:49:10.687347Z
{ "verified": true, "answer": 13108, "timestamp": "2026-02-08T15:49:10.688017Z" }
9fb6d4
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 77, "completion_tokens": 828 }, "timestamp": "2026-02-16T14:42:33.674Z", "answer": 13108 }, { ...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "K3", "status": "ok" }, { "lemma": "K5", "status": "no" }, { "lemma": "V3", "status": "no" } ]
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
e45b57
diophantine_fbi2_count_v1_865884756_541
Let $k$ be the number of ordered pairs $(x_1, x_2)$ of positive odd integers such that $x_1 + x_2 = 960$. Let $m = 170$ and $n = 3$. Determine the number of integers $d$ such that $2 \leq d \leq 170$, $d$ divides $k$, $\frac{k}{d} \geq 3$, and $\frac{k}{d} \leq \sum_{k1=1}^{18} k1$. Compute this number.
20
graphs = [ Graph( let={ "_m": Const(170), "_n": Const(3), "k": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var("x1"), Var("x2")]), condition=And(IsPositive(arg=Var(name='x1')), IsPositive(arg=Var(name='x2')), IsOdd(arg=Var(name='x1')), IsOdd(arg=Var(name='x2')), Eq(...
NT
null
COUNT
sympy
K14
[ "SUM_ARITHMETIC", "COMB1" ]
095a4b
diophantine_fbi2_count_v1
null
5
0
[ "COMB1", "K14", "SUM_ARITHMETIC" ]
3
0.325
2026-02-08T15:29:37.388699Z
{ "verified": true, "answer": 20, "timestamp": "2026-02-08T15:29:37.713991Z" }
b17804
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 162, "completion_tokens": 1594 }, "timestamp": "2026-02-16T07:38:12.423Z", "answer": 20 }, { ...
1
[ { "lemma": "COMB1", "status": "ok" }, { "lemma": "K16", "status": "no" }, { "lemma": "MOD_ADD", "status": "no" }, { "lemma": "SUM_ARITHMETIC", "status": "ok" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", "status": "no" }, { "lemma"...
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
2f70d9
nt_count_phi_equals_v1_865884756_2840
Let $S$ be the set of all ordered pairs $(x, y)$ of positive integers such that $xy = 1562500$. For each such pair, compute $x + y$, and let $u$ be the minimum of these sums. Let $T$ be the set of all integers $t$ such that $5 \leq t \leq 1092$ and there exist positive integers $a$, $b$ with $1 \leq a \leq 315$, $1 \le...
2
graphs = [ Graph( let={ "upper": MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var("x"), Var("y")]), condition=And(IsPositive(arg=Var(name='x')), IsPositive(arg=Var(name='y')), Eq(Mul(Var("x"), Var("y")), Const(1562500)))), expr=Sum(Var("x"), Var("y")))), "k": CountO...
NT
null
COUNT
sympy
LIN_FORM
[ "LIN_FORM", "B3" ]
688dbe
nt_count_phi_equals_v1
null
7
0
[ "B3", "LIN_FORM" ]
2
0.429
2026-02-08T16:58:14.162928Z
{ "verified": true, "answer": 2, "timestamp": "2026-02-08T16:58:14.592020Z" }
698428
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 231, "completion_tokens": 5673 }, "timestamp": "2026-02-17T16:23:42.591Z", "answer": 2 }, { ...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "DS2", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, { "lemma": "MOD_MUL", "status": "no" }, { ...
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
ec1e33
nt_count_divisible_and_v1_168721529_1964
Let $n = 2054$ and let $d_1$ be the number of nonnegative integers $j$ such that $0 \le j \le 2054$ and $\binom{2054}{j}$ is odd. Let $d_2$ be the minimum value of $x + y$ over all ordered pairs $(x, y)$ of positive integers such that $xy = 36$. Compute the number of positive integers $n$ such that $1 \le n \le 25872$,...
1,078
graphs = [ Graph( let={ "_n": Const(2054), "upper": Const(25872), "d1": CountOverSet(set=SolutionsSet(var=Var("j"), condition=And(Geq(Var("j"), Const(0)), Leq(Var("j"), Ref("_n")), Eq(Mod(value=Binom(n=Const(2054), k=Var("j")), modulus=Const(2)), Const(1))), domain='nonne...
ALG
COMB
COUNT
sympy
B3
[ "B3", "V8" ]
5b3848
nt_count_divisible_and_v1
null
6
0
[ "B3", "V8" ]
2
0.863
2026-02-08T14:02:09.418202Z
{ "verified": true, "answer": 1078, "timestamp": "2026-02-08T14:02:10.281320Z" }
b7d641
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 242, "completion_tokens": 617 }, "timestamp": "2026-02-10T00:08:48.433Z", "answer": 1078 }, { "id...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "K15", "status": "no" }, { "lemma": "L3b", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "V3", "status": "no" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", ...
{ "lo": -2.37, "mid": 1.22, "hi": 4.84 }
9dee4d
antilemma_sum_equals_v1_1918700295_3814
Let $T$ be the set of ordered pairs $(i,j)$ such that $1 \leq i \leq 52$, $1 \leq j \leq 53$, and $i + j = 53$. Let $x$ be the number of elements in $T$. Let $S$ be the set of positive integers $t$ such that $16 \leq t \leq 224$ and there exist positive integers $a$, $b$ with $1 \leq a \leq 11$, $1 \leq b \leq 19$, an...
29,306
graphs = [ Graph( let={ "_n": Const(53), "x": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var("i"), Var("j")]), condition=Eq(Sum(Var("i"), Var("j")), Ref("_n")), domain=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(52)), right=IntegerRange(start=Const(1), end=Con...
COMB
GEOM
COMPUTE
sympy
LIN_FORM
[ "LIN_FORM", "COUNT_SUM_EQUALS", "ONE_BINOM_0" ]
4ac49b
antilemma_sum_equals_v1
crt_mix_3
5
0
[ "COUNT_SUM_EQUALS", "LIN_FORM", "ONE_BINOM_0" ]
3
0.008
2026-02-08T08:57:46.064842Z
{ "verified": true, "answer": 29306, "timestamp": "2026-02-08T08:57:46.072620Z" }
f145cf
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 374, "completion_tokens": 30628 }, "timestamp": "2026-02-24T10:15:11.515Z", "answer": 55400 }, { ...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "ok" }, { "lemma": "LIN_F...
{ "lo": 3.25, "mid": 5.68, "hi": 8.81 }
8d62e6
diophantine_sum_product_min_v1_458359167_2229
Let $n = 7$, $S = 8$, and $P = 16$. Define $\text{result}$ to be the smallest positive integer $x$ such that $1 \leq x \leq n$ and $x(S - x) = P$. Let $T$ be the set of all ordered pairs $(x, y)$ of positive integers such that $xy = 1119364$. Define $c$ to be the minimum value of $x + y$ as $(x, y)$ ranges over $T$. Co...
2,112
graphs = [ Graph( let={ "_n": Const(7), "S": Const(8), "P": Const(16), "result": MinOverSet(set=SolutionsSet(var=Var("x"), condition=And(Geq(Var("x"), Const(1)), Leq(Var("x"), Ref("_n")), Eq(Mul(Var("x"), Sub(Ref("S"), Var("x"))), Ref("P"))))), "_c...
NT
null
EXTREMUM
sympy
B3
[ "B3" ]
fc629c
diophantine_sum_product_min_v1
negation_mod
4
0
[ "B3" ]
1
0.004
2026-02-08T05:13:30.470230Z
{ "verified": true, "answer": 2112, "timestamp": "2026-02-08T05:13:30.474680Z" }
5ef26a
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 233, "completion_tokens": 1020 }, "timestamp": "2026-02-11T23:01:51.214Z", "answer": 2112 }, { "...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "MOD_MUL", "status": "no" ...
{ "lo": -3.42, "mid": 1.86, "hi": 7.55 }
8e263b
nt_count_digit_sum_v1_124444284_6857
Let $s$ be the number of integers $t$ such that $7 \leq t \leq 36$ and $t = 3a + 4b$ for some integers $a, b$ with $1 \leq a \leq 4$ and $1 \leq b \leq 6$. Let $r$ be the number of positive integers $n \leq 99999$ such that the sum of the decimal digits of $n$ is equal to $s$. Compute $78961 - r$.
73,086
graphs = [ Graph( let={ "upper": Const(99999), "target_sum": 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)),...
NT
null
COUNT
sympy
LIN_FORM
[ "LIN_FORM" ]
7b2633
nt_count_digit_sum_v1
null
4
0
[ "LIN_FORM" ]
1
3.66
2026-02-08T08:40:39.210699Z
{ "verified": true, "answer": 73086, "timestamp": "2026-02-08T08:40:42.871128Z" }
9fd698
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 156, "completion_tokens": 2499 }, "timestamp": "2026-02-13T20:39:51.841Z", "answer": 73086 }, ...
1
[ { "lemma": "K13", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "MAX_VAL", "status": "no" }, { "lemma": "MOD_MUL", "status": "no" } ]
{ "lo": -3.44, "mid": 1.34, "hi": 6.56 }
c58fc5
nt_count_divisible_v1_1742523217_1128
Let $S$ be the set of all ordered pairs $(x, y)$ of positive integers such that $xy = 49$. Let $d$ be the minimum value of $x + y$ over all pairs $(x, y) \in S$. Determine the number of positive integers $n$ such that $n \leq 66430$ and $n$ is divisible by $d$.
4,745
graphs = [ Graph( let={ "upper": Const(66430), "divisor": 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(49)))), expr=Sum(Var("x"), Var(...
NT
null
COUNT
sympy
B3
[ "B3" ]
0cd20d
nt_count_divisible_v1
null
3
0
[ "B3" ]
1
2.09
2026-02-08T03:26:52.457347Z
{ "verified": true, "answer": 4745, "timestamp": "2026-02-08T03:26:54.547675Z" }
e62a58
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 193, "completion_tokens": 1949 }, "timestamp": "2026-02-10T03:51:01.697Z", "answer": 4745 }, { "i...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "K5", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "MOD_ADD", "status": "no" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, { "lemma": "VAL_SUM_EQ", "status": "no" } ]
{ "lo": -7.01, "mid": -4.68, "hi": -2.15 }
7dbf6b
modular_modexp_compute_v1_1520064083_3884
Let $e$ be the number of positive integers $n$ such that $1 \leq n \leq 19208$ and the $n$th Fibonacci number is divisible by $7$. Compute the remainder when $47^e$ is divided by $18225$.
4,547
graphs = [ Graph( let={ "_n": Const(7), "a": Const(47), "e": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), Const(19208)), Divides(divisor=Ref("_n"), dividend=Fibonacci(arg=Var(name='n')))))), "m": Const(18225)...
NT
null
COMPUTE
sympy
COUNT_FIB_DIVISIBLE
[ "COUNT_FIB_DIVISIBLE" ]
66de3c
modular_modexp_compute_v1
null
6
0
[ "COUNT_FIB_DIVISIBLE" ]
1
0.001
2026-02-08T05:56:53.737349Z
{ "verified": true, "answer": 4547, "timestamp": "2026-02-08T05:56:53.738519Z" }
f9e81c
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 103, "completion_tokens": 3257 }, "timestamp": "2026-02-12T17:29:21.366Z", "answer": 4547 }, {...
1
[ { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "ok" }, { "lemma": "K13", "status": "no" }, { "lemma": "V3", "status": "no" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8_SUM", "status": "no" ...
{ "lo": -3.51, "mid": 1.33, "hi": 6.56 }
54a198
comb_count_permutations_fixed_v1_1520064083_7795
Let $m = 2$. Let $P$ be the set of all ordered pairs $(x,y)$ of positive integers such that $x + y = 10$. Define $n_0$ to be the maximum value of $xy$ over all pairs $(x,y) \in P$. Now, let $Q$ be the set of all ordered pairs $(x,y)$ of positive integers such that $xy = n_0$. Define $n$ to be the minimum value of $x + ...
240
graphs = [ Graph( let={ "_m": Const(2), "_n": MaxOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var("x"), Var("y")]), condition=And(IsPositive(arg=Var(name='x')), IsPositive(arg=Var(name='y')), Eq(Sum(Var("x"), Var("y")), Const(10)))), expr=Mul(Var("x"), Var("y")))), ...
NT
COMB
COUNT
sympy
MIN_PRIME_FACTOR
[ "MIN_PRIME_FACTOR", "B1/B3" ]
57e303
comb_count_permutations_fixed_v1
null
5
0
[ "B1", "B3", "MIN_PRIME_FACTOR" ]
3
0.003
2026-02-08T09:18:31.264391Z
{ "verified": true, "answer": 240, "timestamp": "2026-02-08T09:18:31.267856Z" }
9c42ee
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 223, "completion_tokens": 1005 }, "timestamp": "2026-02-14T03:01:09.321Z", "answer": 240 }, { ...
1
[ { "lemma": "B1", "status": "ok" }, { "lemma": "B3", "status": "ok_later" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "L3b", "status": "no" }, { "lemma": "MIN_PRIME_FACTOR", "status": "ok" }, { "lemma": "V1", "status": "no" }...
{ "lo": -5.14, "mid": 0.32, "hi": 6.51 }
2b03a4
modular_sum_quadratic_residues_v1_1874849503_631
Let $p$ be the largest prime number less than or equal to $521$. Compute $\frac{p(p-1)}{4}$. Multiply this value by $26006$, and let $R$ be the remainder when the product is divided by $85185$. Compute $R$.
16,135
graphs = [ Graph( let={ "_n": Const(85185), "p": MaxOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(2)), Leq(Var("n"), Const(521)), IsPrime(Var("n"))))), "result": Div(Mul(Ref("p"), Sub(Ref("p"), Const(1))), Const(4)), "Q": Mod(value=M...
NT
null
SUM
sympy
MAX_PRIME_BELOW
[ "MAX_PRIME_BELOW" ]
dc3ad3
modular_sum_quadratic_residues_v1
null
4
0
[ "MAX_PRIME_BELOW" ]
1
0.004
2026-02-08T13:13:32.552242Z
{ "verified": true, "answer": 16135, "timestamp": "2026-02-08T13:13:32.556100Z" }
e4a99a
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 170, "completion_tokens": 2295 }, "timestamp": "2026-02-09T19:14:20.674Z", "answer": 16135 }, { "...
1
[ { "lemma": "K14", "status": "no" }, { "lemma": "K15", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok" }, { "lemma": "MOD_SUB", "status": "no" }, { "lemma": "V5", "status": "no" } ]
{ "lo": -5.15, "mid": 0.12, "hi": 6.12 }
3de90e
comb_count_surjections_v1_1978505735_1682
Let $t$ be an integer. Let $s$ be the number of values of $t$ in the range $5 \leq t \leq 22$ for which there exist integers $a$ and $b$ with $1 \leq a \leq 5$, $1 \leq b \leq 4$, and $t = 2a + 3b$. Let $n$ be the number of ordered pairs $(x_1, x_2)$ of positive odd integers such that $x_1 + x_2 = s$. Let $k = 4$. Let ...
41,709
graphs = [ Graph( let={ "_n": Const(65505), "n": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var("x1"), Var("x2")]), condition=And(IsPositive(arg=Var(name='x1')), IsPositive(arg=Var(name='x2')), IsOdd(arg=Var(name='x1')), IsOdd(arg=Var(name='x2')), Eq(Sum(Var("x1"), Var("x2")),...
COMB
null
COUNT
sympy
COMB1
[ "LIN_FORM/COMB1" ]
268a62
comb_count_surjections_v1
null
5
0
[ "COMB1", "LIN_FORM" ]
2
0.031
2026-02-08T16:21:17.173861Z
{ "verified": true, "answer": 41709, "timestamp": "2026-02-08T16:21:17.204570Z" }
eaea07
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 289, "completion_tokens": 3458 }, "timestamp": "2026-02-24T20:37:06.687Z", "answer": 41709 }, { ...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COMB1", "status": "ok_later" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "LIN_FORM"...
{ "lo": -2.4, "mid": 1.57, "hi": 5.75 }
31e0b8
comb_bell_compute_v1_1520064083_5574
Let $T$ be the set of all positive integers $t$ at most 350 for which there exist positive integers $a \leq 21$ and $b \leq 19$ such that $t = 4a + 14b$. Let $n$ be the number of positive integers at most $|T|$ that are divisible by 7 and relatively prime to 6. Let $B_n$ denote the $n$-th Bell number, the number of par...
22,405
graphs = [ Graph( let={ "_m": Const(6), "_n": Const(93527), "n": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), CountOverSet(set=SolutionsSet(var=Var("t"), condition=Exists(var=Var(name='a'), condition=Exists(var=Var(name=...
NT
COMB
COMPUTE
sympy
LIN_FORM
[ "LIN_FORM/C5" ]
683493
comb_bell_compute_v1
null
6
0
[ "C5", "LIN_FORM" ]
2
0.002
2026-02-08T07:26:30.826478Z
{ "verified": true, "answer": 22405, "timestamp": "2026-02-08T07:26:30.828372Z" }
3d0642
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 174, "completion_tokens": 4223 }, "timestamp": "2026-02-13T10:19:37.172Z", "answer": 22405 }, ...
1
[ { "lemma": "C5", "status": "ok_later" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "K18", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "MOD_MUL", "status": "no" }, { "lemma": "MOD_SUB", "status": "no" ...
{ "lo": -3.44, "mid": 1.34, "hi": 6.56 }
06f392
comb_sum_binomial_row_v1_601307018_2466
Let $N$ be the largest positive integer $d$ such that $d^2 \leq 1055744$ and $d \mid 1055744$. Let $M = 2^{15}$. Find the remainder when $N - M$ is divided by $85622$.
53,878
graphs = [ Graph( let={ "n": Const(15), "result": Pow(Const(2), Ref("n")), "_c": MaxOverSet(set=SolutionsSet(var=Var("d"), condition=And(Geq(Var("d"), Const(1)), Divides(divisor=Var("d"), dividend=Const(1055744)), Leq(Mul(Var("d"), Var("d")), Const(1055744))))), ...
COMB
NT
SUM
sympy
B3_CLOSEST
[ "B3_CLOSEST" ]
ff7764
comb_sum_binomial_row_v1
negation_mod
3
0
[ "B3_CLOSEST" ]
1
0.004
2026-03-10T03:12:09.014949Z
{ "verified": true, "answer": 53878, "timestamp": "2026-03-10T03:12:09.018482Z" }
5dcd3c
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 172, "completion_tokens": 32768 }, "timestamp": "2026-03-29T05:27:06.036Z", "answer": null }, { ...
1
[ { "lemma": "B3_CLOSEST", "status": "ok" }, { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "V7", "sta...
{ "lo": 1.23, "mid": 4.27, "hi": 6.7 }
493d35
algebra_poly_eval_v1_1918700295_3387
Let $z$ be the largest prime number $n$ such that $2 \leq n \leq 21$. Compute the value of $$ \frac{12z^3 + 43z^2 + 28z - 18}{85}. $$
1,157
graphs = [ Graph( let={ "_n": Const(3), "z": MaxOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(2)), Leq(Var("n"), Const(21)), IsPrime(Var("n"))))), "result": Div(Sum(Mul(Const(12), Pow(Ref("z"), Ref("_n"))), Mul(Const(43), Pow(Ref("z"), Const(2))...
NT
null
COMPUTE
sympy
MAX_PRIME_BELOW
[ "MAX_PRIME_BELOW" ]
dc3ad3
algebra_poly_eval_v1
null
2
0
[ "MAX_PRIME_BELOW" ]
1
0.004
2026-02-08T08:35:50.887391Z
{ "verified": true, "answer": 1157, "timestamp": "2026-02-08T08:35:50.891335Z" }
30cfbf
CC BY 4.0
[ { "id": 8, "model": "mathstral", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 116, "completion_tokens": 430 }, "timestamp": "2026-02-15T20:16:55.237Z", "answer": 1149 }, { "id": 11,...
1
[ { "lemma": "K18", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok" }, { "lemma": "V3", "status": "no" }, { "lemma": "V5", "status": "no" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", "status": "no" } ]
{ "lo": -7.01, "mid": -4.68, "hi": -2.15 }
f61eaa
modular_inverse_v1_397696148_1995
Let $a = 178$. Let $m$ be the number of positive integers $n$ such that $1 \leq n \leq 1891$ and $\gcd(n, 10) = 1$. Let $S$ be the set of all ordered pairs $(x, y)$ of positive integers such that $xy = 142884$. Define $s_{\text{min}}$ to be the minimum value of $x + y$ as $(x, y)$ ranges over $S$. Determine the smalles...
370
graphs = [ Graph( let={ "a": Const(178), "m": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), Const(1891)), Eq(GCD(a=Var("n"), b=Const(10)), Const(1))))), "upper": MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elemen...
NT
null
EXTREMUM
sympy
B3
[ "B3", "C4" ]
8d18b3
modular_inverse_v1
null
6
0
[ "B3", "C4" ]
2
0.035
2026-02-08T12:53:48.707482Z
{ "verified": true, "answer": 370, "timestamp": "2026-02-08T12:53:48.742677Z" }
d11278
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 186, "completion_tokens": 1599 }, "timestamp": "2026-02-15T06:46:00.923Z", "answer": 370 }, { ...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "C4", "status": "ok" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "L3b", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no"...
{ "lo": -5.14, "mid": 0.32, "hi": 6.51 }
94c45c
geo_count_lattice_triangle_v1_1218484723_46
Let $M = \left|\min\{ 9b^3 + 51a^2b + 35a^3 + 33ab^2 \mid 1 \le a, b \le 14\} \cdot 127 + 41 \cdot (-128)\right|$, and let $R = \gcd(128, 128) + \gcd(|41 - 128|, |127 - 128|) + \gcd(|0 - 41|, |0 - 127|)$. Compute $\frac{M + 2 - R}{2}$.
5,440
graphs = [ Graph( let={ "_n": Const(127), "area_2x": Abs(arg=Sum(Mul(MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var(name='a'), Var(name='b')]), condition=And(Geq(left=Var(name='a'), right=Const(value=1)), Leq(left=Var(name='a'), right=Const(value=14)), Geq(left=Va...
GEOM
NT
COUNT
sympy
POLY3_MIN
[ "POLY3_MIN" ]
e2e279
geo_count_lattice_triangle_v1
null
7
0
[ "POLY3_MIN" ]
1
0.005
2026-02-25T01:44:37.298039Z
{ "verified": true, "answer": 5440, "timestamp": "2026-02-25T01:44:37.302975Z" }
186795
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 241, "completion_tokens": 905 }, "timestamp": "2026-03-28T21:37:18.660Z", "answer": 5440 }, { "id...
1
[ { "lemma": "POLY3_MIN", "status": "ok" } ]
{ "lo": -0.93, "mid": 2.04, "hi": 4.6 }
fa9f92
modular_count_residue_v1_717093673_3762
Let $A$ be the set of all ordered pairs $(x, y)$ of positive integers such that $xy = 6718464$. Let $s$ be the minimum value of $x + y$ as $(x, y)$ ranges over $A$. Let $B$ be the set of all ordered pairs $(x_1, y_1)$ of positive integers such that $x_1 y_1 = s$. Let $r$ be the minimum value of $x_1 + y_1$ as $(x_1, y_...
1,170
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")), Const(6718464)))), expr=Sum(Var("x"), Var("y")))), "upper": Const...
NT
null
COUNT
sympy
B3
[ "B3/B3" ]
8ffef9
modular_count_residue_v1
null
6
0
[ "B3" ]
1
2.116
2026-02-08T17:49:49.754774Z
{ "verified": true, "answer": 1170, "timestamp": "2026-02-08T17:49:51.870815Z" }
3e845b
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 203, "completion_tokens": 1890 }, "timestamp": "2026-02-18T08:57:03.690Z", "answer": 1170 }, {...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "MAX_VAL", "status": "no" }, { "lemma": "V1", "status": "no" } ]
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
76102b
antilemma_sum_equals_v1_1978505735_4
Let $n$ be the number of integers $t$ such that $8 \leq t \leq 60$ and there exist positive integers $a$ and $b$ with $1 \leq a \leq 5$, $1 \leq b \leq 9$, and $t = 3a + 5b$. Let $x$ be the number of ordered pairs $(i, j)$ such that $1 \leq i \leq 43$, $1 \leq j \leq 43$, and $i + j = n$. Compute $77777 - x$.
77,735
graphs = [ Graph( let={ "_m": Const(77777), "_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=5)), Geq(left=V...
COMB
GEOM
COMPUTE
sympy
LIN_FORM
[ "LIN_FORM/COUNT_SUM_EQUALS", "COUNT_SUM_EQUALS" ]
ada383
antilemma_sum_equals_v1
null
4
0
[ "COUNT_SUM_EQUALS", "LIN_FORM" ]
2
0.072
2026-02-08T15:08:19.361617Z
{ "verified": true, "answer": 77735, "timestamp": "2026-02-08T15:08:19.433130Z" }
3b8cf0
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 241, "completion_tokens": 3897 }, "timestamp": "2026-02-10T06:39:11.868Z", "answer": 77735 }, { "...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "ok" }, { "lemma": "LIN_F...
{ "lo": -2.43, "mid": 1.2, "hi": 4.84 }
a098d2
comb_bell_compute_v1_1116507919_238
Let $n$ be the number of ordered pairs $(i, j)$ such that $1 \leq i \leq 2$ and $1 \leq j \leq 4$. Let $B_n$ denote the $n$-th Bell number, which counts the number of partitions of a set of $n$ elements. Compute the remainder when $11201 \cdot B_n$ is divided by 84192.
66,540
graphs = [ Graph( let={ "n": CountOverSet(set=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(2)), right=IntegerRange(start=Const(1), end=Const(4)))), "result": Bell(Ref("n")), "_c": Const(11201), "Q": Mod(value=Mul(Ref("_c"), Ref("result")), modu...
COMB
null
COMPUTE
sympy
COUNT_CARTESIAN
[ "COUNT_CARTESIAN" ]
409d7d
comb_bell_compute_v1
null
4
0
[ "COUNT_CARTESIAN" ]
1
0.001
2026-02-08T02:29:32.923834Z
{ "verified": true, "answer": 66540, "timestamp": "2026-02-08T02:29:32.925016Z" }
7c4dd7
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 206, "completion_tokens": 1069 }, "timestamp": "2026-02-08T19:16:27.827Z", "answer": 66540 }, { "...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_CARTESIAN", "status": "ok" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8_SUM", "statu...
{ "lo": -0.94, "mid": 0.81, "hi": 2.32 }
f915b7
nt_count_gcd_equals_v1_151522320_2292
Let $S$ be the set of all ordered pairs $(x, y)$ of positive integers such that $x + y = 158$. Let $u$ be the maximum value of $xy$ over all such pairs. Let $r$ be the number of positive integers $n$ with $1 \leq n \leq u$ such that $\gcd(n, 351) = 1$. Let $p_{\text{max}}$ be the largest prime number $n$ such that $2 \...
31,570
graphs = [ Graph( let={ "_m": Const(2), "_n": Const(313), "upper": MaxOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var("x"), Var("y")]), condition=And(IsPositive(arg=Var(name='x')), IsPositive(arg=Var(name='y')), Eq(Sum(Var("x"), Var("y")), Const(158)))), e...
NT
null
COUNT
sympy
MAX_PRIME_BELOW
[ "MAX_PRIME_BELOW", "B1" ]
490d53
nt_count_gcd_equals_v1
two_moduli
5
0
[ "B1", "MAX_PRIME_BELOW" ]
2
3.836
2026-02-08T04:43:42.237189Z
{ "verified": true, "answer": 31570, "timestamp": "2026-02-08T04:43:46.073558Z" }
e99a0b
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 263, "completion_tokens": 1595 }, "timestamp": "2026-02-11T21:48:58.224Z", "answer": 31670 }, { ...
1
[ { "lemma": "B1", "status": "ok" }, { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, { "lemma": "POLY_PADIC_VAL_CONST", ...
{ "lo": 1.71, "mid": 5.04, "hi": 8.38 }
0282e9
modular_min_modexp_v1_1439011603_3090
Let $a = 7$. Let $b$ be the number of integers $t$ such that $12 \leq t \leq 462$ and there exist integers $a'$ and $b'$ satisfying $1 \leq a' \leq 61$, $1 \leq b' \leq 7$, and $t = 7a' + 5b'$. Let $m = 563$. Determine the value of the smallest positive integer $x$ such that $1 \leq x \leq 281$ and $7^x \equiv b \pmod{...
143
graphs = [ Graph( let={ "a": Const(7), "b": 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=61)), Geq(left=Var(na...
NT
null
EXTREMUM
sympy
LIN_FORM
[ "LIN_FORM" ]
7b2633
modular_min_modexp_v1
null
5
0
[ "LIN_FORM" ]
1
0.03
2026-02-08T17:13:58.307814Z
{ "verified": true, "answer": 143, "timestamp": "2026-02-08T17:13:58.338311Z" }
a6de89
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 171, "completion_tokens": 8013 }, "timestamp": "2026-02-17T22:12:09.450Z", "answer": 143 }, { ...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "K14", "status": "no" }, { "lemma": "K15", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "MAX_VAL", "status": "no" } ]
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
533f7d
antilemma_k2_v1_168721529_981
Let $x = \sum_{k=1}^{271} \phi(k) \left\lfloor \frac{271}{k} \right\rfloor$, where $\phi(k)$ denotes Euler's totient function. Let $c = 46165$. Compute the remainder when $c \cdot x$ is divided by $92489$.
29,596
graphs = [ Graph( let={ "x": Summation(var="k", start=Const(1), end=Const(271), expr=Mul(EulerPhi(n=Var("k")), Floor(Div(Const(271), Var("k"))))), "_c": Const(46165), "Q": Mod(value=Mul(Ref("_c"), Ref("x")), modulus=Const(92489)), }, goal=Ref("Q"), ) ]
NT
COMB
COMPUTE
sympy
K2
[ "K2" ]
6897ab
antilemma_k2_v1
null
4
0
[ "K2" ]
1
0.001
2026-02-08T13:22:23.959024Z
{ "verified": true, "answer": 29596, "timestamp": "2026-02-08T13:22:23.960300Z" }
21ca99
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 187, "completion_tokens": 3427 }, "timestamp": "2026-02-09T11:32:38.468Z", "answer": 29596 }, { "...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "DS2", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "K2", "status": "ok" }, { "lemma": "L3c", "status": "no" }, { "lemma": "VAL_SUM_EQ", "status": "no" } ]
{ "lo": -1.95, "mid": 2.14, "hi": 6.33 }
b5d2fb
diophantine_fbi2_min_v1_1742523217_4622
Let $S$ be the set of all ordered pairs $(x, y)$ of positive integers such that $xy = 49$. Let $k$ be the minimum value of $x + y$ over all such pairs. Let $d$ be a positive integer such that $2 \leq d \leq 24$, $d$ divides $k$, and $\frac{k}{d} \geq 4$. Let $r$ be the smallest such $d$. Compute the remainder when $987...
29,030
graphs = [ Graph( let={ "_n": Const(84204), "k": 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(49)))), expr=Sum(Var("x"), Var("y")))), ...
NT
null
EXTREMUM
sympy
B3
[ "B3" ]
0cd20d
diophantine_fbi2_min_v1
null
4
0
[ "B3" ]
1
0.005
2026-02-08T08:59:54.123790Z
{ "verified": true, "answer": 29030, "timestamp": "2026-02-08T08:59:54.128506Z" }
d6fa04
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 161, "completion_tokens": 628 }, "timestamp": "2026-02-13T23:09:33.155Z", "answer": 29030 }, {...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "L3b", "status": "no" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, { "lemma": "POLY_PADIC_VAL_CONST", ...
{ "lo": -5.55, "mid": -3.01, "hi": 0.32 }
3a8e24
algebra_quadratic_discriminant_v1_458359167_5801
Let $m = 2$. Let $A$ be the set of all positive integers $p$ for which there exists a positive integer $q$ such that $pq = 900$, $\gcd(p, q) = 1$, and $p < q$. Let $T$ be the number of elements in $A$. Let $E$ be the set of all positive integers $n$ such that $1 \leq n \leq m$ and $n$ is even. Let $s$ be the sum of all...
31,280
graphs = [ Graph( let={ "_m": Const(2), "_n": Const(61790), "a": Const(-2), "b": Const(-10), "c": Const(-5), "result": Sub(Pow(Ref("b"), SumOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), Ref(...
NT
null
COMPUTE
sympy
COPRIME_PAIRS
[ "COPRIME_PAIRS", "SUM_DIVISIBLE" ]
e5b6b4
algebra_quadratic_discriminant_v1
null
3
0
[ "COPRIME_PAIRS", "SUM_DIVISIBLE" ]
2
0.004
2026-02-08T12:42:04.835293Z
{ "verified": true, "answer": 31280, "timestamp": "2026-02-08T12:42:04.839278Z" }
42d6e3
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 220, "completion_tokens": 1379 }, "timestamp": "2026-02-15T03:57:19.289Z", "answer": 31280 }, ...
1
[ { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, { "lemma": "SUM_DIVISIBLE", "status": "ok" }, { "lemma": "V8_SUM", "stat...
{ "lo": -5.14, "mid": 0.32, "hi": 6.51 }
fe0b45
sequence_fibonacci_compute_v1_2051736721_5981
Let $n$ be the number of nonnegative integers $j$ with $0 \leq j \leq 16451$ such that $\binom{16451}{j}$ is odd, increased by $4$. Compute the $n$-th Fibonacci number, where the Fibonacci sequence is defined by $F_1 = 1$, $F_2 = 1$, and $F_k = F_{k-1} + F_{k-2}$ for $k \geq 3$.
6,765
graphs = [ Graph( let={ "_n": Const(2), "n": Sum(CountOverSet(set=SolutionsSet(var=Var("j"), condition=And(Geq(Var("j"), Const(0)), Leq(Var("j"), Const(16451)), Eq(Mod(value=Binom(n=Const(16451), k=Var("j")), modulus=Ref("_n")), Const(1))), domain='nonnegative_integers')), Const(4)),...
ALG
COMB
COMPUTE
sympy
V8
[ "V8" ]
86348e
sequence_fibonacci_compute_v1
null
7
0
[ "V8" ]
1
0.002
2026-02-08T18:53:41.327182Z
{ "verified": true, "answer": 6765, "timestamp": "2026-02-08T18:53:41.328891Z" }
53089a
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 149, "completion_tokens": 1748 }, "timestamp": "2026-02-18T20:15:53.368Z", "answer": 6765 }, {...
1
[ { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", "status": "...
{ "lo": -3.12, "mid": 1.47, "hi": 6.57 }