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f964d0
nt_count_divisors_in_range_v1_1353956133_26
Let $a = 1$ and $n = 27720$. Let $b$ be the number of positive integers $m$ such that $1 \leq m \leq 60683$, $7$ divides $m$, and $\gcd(m, 10) = 1$. Let $c$ be the number of positive divisors $d$ of $n$ such that $a \leq d \leq b$. Find the remainder when $25849 \cdot c$ is divided by $75954$.
21,941
graphs = [ Graph( let={ "_n": Const(10), "n": Const(27720), "a": Const(1), "b": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), Const(60683)), Divides(divisor=Const(7), dividend=Var("n")), Eq(GCD(a=Var("n"), b=R...
NT
null
COUNT
sympy
C5
[ "C5" ]
1d9668
nt_count_divisors_in_range_v1
null
5
0
[ "C5" ]
1
0.035
2026-02-08T11:16:29.982510Z
{ "verified": true, "answer": 21941, "timestamp": "2026-02-08T11:16:30.017895Z" }
a1dddc
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 153, "completion_tokens": 2562 }, "timestamp": "2026-02-14T11:24:49.488Z", "answer": 21941 }, ...
1
[ { "lemma": "C5", "status": "ok" }, { "lemma": "K17", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, { "lemma": "MOD_MUL", "status": "no" }, { "lemma": "MOD_SUB", "status": "no" } ]
{ "lo": -5.14, "mid": 0.32, "hi": 6.51 }
3d31a3
modular_product_range_v1_601307018_9235
Let $S = \left\{ v \mid 4 \le v \le 2624,\ \exists\, a,b \in \mathbb{Z},\ 1 \le a,b \le 11\ \text{such that}\ 25a^2 - 38ab + 17b^2 = v \right\}$. Let $M = \prod_{i=4}^{|S|} i$. Find the remainder when $M$ is divided by $10427$.
6,123
graphs = [ Graph( let={ "_n": Const(4), "prod": MathProduct(expr=Var("i"), var="i", start=Const(4), end=CountOverSet(set=SolutionsSet(var=Var("v"), condition=And(Geq(Var("v"), Ref("_n")), Leq(Var("v"), Const(2624)), Exists(var=Tuple(elements=[Var(name='a'), Var(name='b')]), condition...
NT
null
COMPUTE
sympy
QF_PSD_DISTINCT
[ "QF_PSD_DISTINCT" ]
a8f9cb
modular_product_range_v1
null
4
0
[ "QF_PSD_DISTINCT" ]
1
0.008
2026-03-10T09:36:53.741828Z
{ "verified": true, "answer": 6123, "timestamp": "2026-03-10T09:36:53.749673Z" }
3c92d1
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 216, "completion_tokens": 22852 }, "timestamp": "2026-04-19T10:55:59.449Z", "answer": 6123 }, { ...
1
[ { "lemma": "DS2", "status": "no" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, { "lemma": "MOD_MUL", "status": "no" }, { "lemma": "QF_PSD_DISTINCT", "status": "ok" }, { "lemma": "V1", "status": "no" }, { "lemma": "V5", "status": "no" } ]
{ "lo": -5.34, "mid": 1.1, "hi": 7.53 }
586252
geo_count_lattice_triangle_v1_1419126231_331
Let $M = \left|100 \cdot 144 + 12 \cdot (0 - 300)\right|$. Let $R = \gcd\left(100, \sum_{\substack{(a, b, c),\, a^2 + b^2 + c^2 = ab + bc + ca,\\ 4a + 3b + 5c = 120,\, a \geq 1, b \geq 1, c \geq 1}} (a^2 + b^2 + c^2)\right) + \gcd(|12 - 100|, |144 - 300|) + \gcd(|0 - 12|, |0 - 144|)$. Let $S = \frac{M + 2 - R}{2}$. Com...
57,157
graphs = [ Graph( let={ "_n": Const(2), "area_2x": Abs(arg=Sum(Mul(Const(value=100), Const(value=144)), Mul(Const(value=12), Sub(left=Const(value=0), right=Const(value=300))))), "boundary": Sum(GCD(a=Abs(arg=Const(value=100)), b=Abs(arg=SumOverSet(set=MapOverSet(set=Solut...
GEOM
NT
COUNT
sympy
SUM_SQUARES_IDENTITY
[ "SUM_SQUARES_IDENTITY" ]
9879b8
geo_count_lattice_triangle_v1
null
5
0
[ "SUM_SQUARES_IDENTITY" ]
1
0.006
2026-02-25T09:50:40.042462Z
{ "verified": true, "answer": 57157, "timestamp": "2026-02-25T09:50:40.048690Z" }
b5f9b0
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 296, "completion_tokens": 1101 }, "timestamp": "2026-03-30T08:04:28.855Z", "answer": 57157 }, { "...
1
[ { "lemma": "SUM_SQUARES_IDENTITY", "status": "ok" } ]
{ "lo": -5.37, "mid": 0.23, "hi": 5.22 }
dde0b5
nt_min_coprime_above_v1_1248542787_150
Let $A$ be the set of all ordered pairs $(i,j)$ where $i$ is an integer from 1 to 20, $j$ is an integer from 1 to 29, and $\gcd(i,j) = 1$. Let $m$ be the number of elements in $A$. Let $n_0$ be the smallest integer $n$ such that $85849 < n \leq 86230$ and $\gcd(n, m) = 1$. Compute the remainder when $\sum_{k=1}^{n_0} \...
41,263
graphs = [ Graph( let={ "start": Const(85849), "upper": Const(86230), "modulus": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var("i"), Var("j")]), condition=Eq(GCD(a=Var("i"), b=Var("j")), Const(1)), domain=CartesianProduct(left=IntegerRange(start=Const(1), end=Cons...
NT
null
EXTREMUM
sympy
COUNT_COPRIME_GRID
[ "COUNT_COPRIME_GRID" ]
20ec03
nt_min_coprime_above_v1
null
6
0
[ "COUNT_COPRIME_GRID" ]
1
0.043
2026-02-08T02:58:21.432795Z
{ "verified": true, "answer": 41263, "timestamp": "2026-02-08T02:58:21.475411Z" }
de7196
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 257, "completion_tokens": 32768 }, "timestamp": "2026-02-23T21:15:58.945Z", "answer": null }, { ...
0
[ { "lemma": "COUNT_COPRIME_GRID", "status": "ok" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "MOD_ADD", "status": "no" }, { "lemma": "V5", "status": "no" }, { "lemma": "VAL_SUM_EQ", ...
{ "lo": 4.56, "mid": 6.51, "hi": 9.5 }
2c9252
nt_max_prime_below_v1_1915831931_3920
Let $S$ be the set of all positive integers $p$ such that there exists a positive integer $q$ with $pq = 12$, $\gcd(p, q) = 1$, and $p < q$. Let $c$ be the number of elements in $S$. Let $n$ be a prime number satisfying $c \leq n \leq 80656$. Determine the value of the largest such $n$.
80,651
graphs = [ Graph( let={ "upper": Const(80656), "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
1.923
2026-02-08T18:01:06.375685Z
{ "verified": true, "answer": 80651, "timestamp": "2026-02-08T18:01:08.298250Z" }
f35fc7
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": 4507 }, "timestamp": "2026-02-18T11:59:41.015Z", "answer": 80651 }, ...
1
[ { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "L3c", "status": "no" }, { "lemma": "MOD_ADD", "status": "no" }, { "lemma": "V8_SUM", "s...
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
d2ba81
nt_count_divisible_v1_677425708_2170
Let $n = 2$. Define $T$ as the set of all integers $t$ such that $7 \leq t \leq 20$ and there exist positive integers $a \leq 5$ and $b \leq 2$ satisfying $t = 2a + 5b$. Let $d$ be the largest prime number $n$ such that $n \geq 2$ and $n \leq |T|$. Define $r$ as the number of positive integers $m$ such that $1 \leq m \...
7,296
graphs = [ Graph( let={ "_n": Const(2), "upper": Const(51076), "divisor": MaxOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Ref("_n")), Leq(Var("n"), CountOverSet(set=SolutionsSet(var=Var("t"), condition=Exists(var=Var(name='a'), condition=Exists(var=V...
NT
null
COUNT
sympy
LIN_FORM
[ "LIN_FORM/MAX_PRIME_BELOW", "ONE_PHI_1" ]
fc144b
nt_count_divisible_v1
null
4
0
[ "LIN_FORM", "MAX_PRIME_BELOW", "ONE_PHI_1" ]
3
3.489
2026-02-08T04:49:54.859676Z
{ "verified": true, "answer": 7296, "timestamp": "2026-02-08T04:49:58.348660Z" }
beea66
CC BY 4.0
[ { "id": 8, "model": "mathstral", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 234, "completion_tokens": 600 }, "timestamp": "2026-02-18T14:03:39.415Z", "answer": 7296 } ]
2
[ { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok_later" }, { "lemma": "MAX_VAL", "status": "no" }, { "lemma": "ONE_PHI_1", "s...
{ "lo": -10, "mid": -6.47, "hi": -2.95 }
4d64a7
comb_sum_binomial_row_v1_784195855_7684
Let $\_n = 2$. Let $n$ be the number of positive integers $p$ for which there exists a positive integer $q$ such that $p \cdot q = 11205810$, $\gcd(p, q) = 1$, and $p < q$. Compute $\_n^n$.
65,536
graphs = [ Graph( let={ "_n": Const(2), "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=11205810)), Eq(left=GCD(a=Var(name='p'), b=Var(na...
NT
null
SUM
sympy
COPRIME_PAIRS
[ "COPRIME_PAIRS" ]
2bb3aa
comb_sum_binomial_row_v1
null
3
0
[ "COPRIME_PAIRS" ]
1
0.001
2026-02-08T09:26:42.836500Z
{ "verified": true, "answer": 65536, "timestamp": "2026-02-08T09:26:42.837982Z" }
9fca18
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 113, "completion_tokens": 1507 }, "timestamp": "2026-02-14T04:20:50.282Z", "answer": 65536 }, ...
1
[ { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "K13", "status": "no" }, { "lemma": "K15", "status": "no" }, { "lemma": "K18", "status": "no" }, { "lemma": "L3b", "status": "no" } ]
{ "lo": -5.14, "mid": 0.32, "hi": 6.51 }
25a1b4
diophantine_sum_product_min_v1_1742523217_952
Let $c=2$ and $m=2$. Consider all integers $x$ such that $$x^2-102x-208=0.$$ Let $S_0$ be the set of all such integers $x$, and let $$T=\sum_{x\in S_0} x.$$ Let $n$ be the largest prime number with $2\le n\le T$. Let $A$ be the set of all integers $t$ such that there exist integers $a$ and $b$ with $1\le a\le 10$, $1...
39
graphs = [ Graph( let={ "_c": Const(2), "_m": Const(2), "_n": MaxOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Ref("_m")), Leq(Var("n"), SumOverSet(set=SolutionsSet(var=Var("x"), condition=Eq(Sum(Pow(Var("x"), Ref("_c")), Mul(Const(-102), Var("x")), C...
NT
null
EXTREMUM
sympy
K13
[ "VIETA_SUM/MAX_PRIME_BELOW/LIN_FORM" ]
65772e
diophantine_sum_product_min_v1
null
8
0
[ "K13", "LIN_FORM", "MAX_PRIME_BELOW", "VIETA_SUM" ]
4
0.484
2026-02-08T03:22:02.911995Z
{ "verified": true, "answer": 39, "timestamp": "2026-02-08T03:22:03.395981Z" }
a644b4
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 425, "completion_tokens": 20343 }, "timestamp": "2026-02-23T18:18:39.905Z", "answer": 39 }, { "i...
1
[ { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "K13", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok_later" }, { "lemma": "MAX_PRIME_BELOW", "status": "o...
{ "lo": 2.84, "mid": 4.91, "hi": 7.14 }
255394
antilemma_k2_v1_1915831931_3169
Let $x = \sum_{k=1}^{66} \phi(k) \left\lfloor \frac{66}{k} \right\rfloor$, where $\phi$ denotes Euler's totient function. Let $c = 62648$. Compute the remainder when $c \cdot x$ is divided by 83733. Find the value of $Q$, where $Q$ is this remainder.
20,346
graphs = [ Graph( let={ "_n": Const(66), "x": Summation(var="k", start=Const(1), end=Ref("_n"), expr=Mul(EulerPhi(n=Var("k")), Floor(Div(Const(66), Var("k"))))), "_c": Const(62648), "Q": Mod(value=Mul(Ref("_c"), Ref("x")), modulus=Const(83733)), }, ...
NT
COMB
COMPUTE
sympy
K2
[ "K2" ]
6897ab
antilemma_k2_v1
null
5
0
[ "K2" ]
1
0.001
2026-02-08T17:24:26.169159Z
{ "verified": true, "answer": 20346, "timestamp": "2026-02-08T17:24:26.170149Z" }
a254cc
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 130, "completion_tokens": 2471 }, "timestamp": "2026-02-18T01:11:57.587Z", "answer": 20346 }, ...
1
[ { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "K13", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "K2", "status": "ok" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "MAX_VAL", "status": "no" } ]
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
2eaa35
algebra_poly_eval_v1_124444284_907
Let $n = 3$ and $x = 10$. Define $r = 5x^n + 7x^2 + 7x - 3$. Let $c$ be the number of integers $t$ such that $7 \leq t \leq 2411$ and there exist positive integers $a$, $b$ with $1 \leq a \leq 321$, $1 \leq b \leq 403$, and $t = 5a + 2b$. Compute the remainder when $r^2 + 47r + c$ is divided by $61269$.
17,596
graphs = [ Graph( let={ "_n": Const(3), "x": Const(10), "result": Sum(Mul(Const(5), Pow(Ref("x"), Ref("_n"))), Mul(Const(7), Pow(Ref("x"), Const(2))), Mul(Const(7), Ref("x")), Const(-3)), "_c": CountOverSet(set=SolutionsSet(var=Var("t"), condition=Exists(var=V...
NT
null
COMPUTE
sympy
LIN_FORM
[ "LIN_FORM" ]
2ba0ea
algebra_poly_eval_v1
quadratic_mod
5
0
[ "LIN_FORM" ]
1
0.002
2026-02-08T03:35:45.528862Z
{ "verified": true, "answer": 17596, "timestamp": "2026-02-08T03:35:45.531030Z" }
077dc1
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 244, "completion_tokens": 2150 }, "timestamp": "2026-02-09T23:43:40.753Z", "answer": 17600 }, { ...
1
[ { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "MOD_MUL", "status": "no" }, { "lemma": "MOD_SUB", "status": "no" }, { "lemma": "V5", "status...
{ "lo": 1.71, "mid": 5.04, "hi": 8.38 }
d7fcc4
diophantine_sum_product_min_v1_1520064083_5769
Let $S=45$. Consider all ordered pairs $(x,y)$ of positive integers such that $xy=63504$. Let $P$ be the minimum possible value of $x+y$ over all such pairs. Let $T$ be the set of all integers $z$ such that $$z^2-484z+34848=0.$$ Let $U$ be the sum of all elements of $T$. Consider all ordered pairs $(u,v)$ of positiv...
86,121
graphs = [ Graph( let={ "_n": Const(21505), "S": Const(45), "P": MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var("x"), Var("y")]), condition=And(IsPositive(arg=Var(name='x')), IsPositive(arg=Var(name='y')), Eq(Mul(Var("x"), Var("y")), Const(63504)))), e...
NT
null
EXTREMUM
sympy
VIETA_SUM
[ "VIETA_SUM/B3" ]
d036a4
diophantine_sum_product_min_v1
null
8
0
[ "B3", "VIETA_SUM" ]
2
0.019
2026-02-08T07:37:18.840446Z
{ "verified": true, "answer": 86121, "timestamp": "2026-02-08T07:37:18.859716Z" }
f7b3ed
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 272, "completion_tokens": 1699 }, "timestamp": "2026-02-13T11:11:46.143Z", "answer": 86121 }, ...
1
[ { "lemma": "B3", "status": "ok_later" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "DS2", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "MAX_VAL", "status": "no" }, { "lemma": "MOD_MUL", "status": "no" }, ...
{ "lo": -3.51, "mid": 1.33, "hi": 6.56 }
6c8b84
algebra_quadratic_discriminant_v1_655260480_407
Let $a = 1$, $b = 11$, and $c = 18$. Define the discriminant $D = b^2 - 4ac$. Let $r = 2$ if $D > 0$, $r = 1$ if $D = 0$, and $r = 0$ otherwise. Compute the Bell number $B_k$, where $k = |r| \bmod 11$.
2
graphs = [ Graph( let={ "a": Const(1), "b": Const(11), "c": Const(18), "D": Sub(Pow(Ref("b"), Const(2)), Mul(Const(4), Ref("a"), Ref("c"))), "result": Sum(Mul(Const(2), Iverson(condition=Gt(Ref("D"), Const(0)))), Iverson(condition=Eq(Ref("D"), Cons...
COMB
null
COMPUTE
sympy
MAX_PRIME_BELOW
[ "B3/L3C" ]
345f3b
algebra_quadratic_discriminant_v1
null
2
0
[ "B3", "L3C", "MAX_PRIME_BELOW" ]
3
0.036
2026-02-08T15:22:14.698158Z
{ "verified": true, "answer": 2, "timestamp": "2026-02-08T15:22:14.734156Z" }
e9a784
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": 289 }, "timestamp": "2026-02-24T20:38:36.923Z", "answer": 2 }, { "id": ...
1
[ { "lemma": "B3", "status": "ok" }, { "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", ...
{ "lo": -8.52, "mid": -5.43, "hi": -3.2 }
cb1e28_l
antilemma_sum_equals_v1_124444284_2022
Let $x$ be the number of ordered pairs $(i,j)$ of positive integers with $1 \leq i \leq 56$, $1 \leq j \leq 56$, and $i + j = 58$. Compute the remainder when $12 - x$ is divided by $53960$.
53,916
COMB
GEOM
COMPUTE
sympy
COUNT_SUM_EQUALS
[ "COUNT_SUM_EQUALS" ]
75ab0f
antilemma_sum_equals_v1
null
2
0
[ "COUNT_SUM_EQUALS" ]
1
0.006
2026-02-08T04:15:32.376915Z
{ "verified": false, "answer": 53917, "timestamp": "2026-02-08T04:15:32.382692Z" }
24b2e3
cb1e28
legacy_text
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 177, "completion_tokens": 764 }, "timestamp": "2026-02-23T23:56:25.261Z", "answer": 53917 }, { "i...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "ok" }, { "lemma": "V8", "status": "no" }, { "lemma": "V8_SUM", "status"...
{ "lo": -3.83, "mid": -1.68, "hi": 1.09 }
3642ef
nt_count_divisible_v1_1520064083_3801
Let $ S $ be the sum $$ S = \sum_{k=0}^{9} (-1)^k \binom{9}{k}. $$ Let $ A $ be the set of all positive integers $ n $ such that $ 1 \leq n \leq 39204 $ and $ n \equiv S \pmod{19} $. Compute $ 10731 $ minus the number of elements in $ A $.
8,668
graphs = [ Graph( let={ "upper": Const(39204), "divisor": Const(19), "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("divisor")), Summation(var="k", start=Const(0)...
COMB
null
COUNT
sympy
BINOMIAL_ALTERNATING
[ "BINOMIAL_ALTERNATING" ]
c21569
nt_count_divisible_v1
null
5
0
[ "BINOMIAL_ALTERNATING" ]
1
1.27
2026-02-08T05:53:01.258768Z
{ "verified": true, "answer": 8668, "timestamp": "2026-02-08T05:53:02.528370Z" }
a46e2a
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 204, "completion_tokens": 707 }, "timestamp": "2026-02-24T04:50:08.181Z", "answer": 8668 }, { "id...
1
[ { "lemma": "BINOMIAL_ALTERNATING", "status": "ok" }, { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", "status": "no" }, { "lemma": "V8_SUM", "stat...
{ "lo": -5.98, "mid": -3.97, "hi": -1.94 }
cced62
comb_count_derangements_v1_1915831931_1635
Let $m = 93505$. Let $d_{\text{min}}$ be the smallest divisor of $437$ that is at least $2$. Let $n$ be the number of prime numbers less than or equal to $d_{\text{min}}$ and at least $2$. Compute the remainder when $83872 \cdot !n$ is divided by $m$, where $!n$ denotes the subfactorial of $n$. Enter your answer as an ...
82,856
graphs = [ Graph( let={ "_m": Const(93505), "_n": MinOverSet(set=SolutionsSet(var=Var("d"), condition=And(Geq(Var("d"), Const(2)), Divides(divisor=Var("d"), dividend=Const(437))))), "n": CountOverSet(set=SolutionsSet(var=Var("n1"), condition=And(Geq(Var("n1"), Const(2)), ...
NT
COMB
COUNT
sympy
MIN_PRIME_FACTOR
[ "MIN_PRIME_FACTOR/COUNT_PRIMES" ]
56ea03
comb_count_derangements_v1
null
4
0
[ "COUNT_PRIMES", "MIN_PRIME_FACTOR" ]
2
0.002
2026-02-08T16:19:00.765795Z
{ "verified": true, "answer": 82856, "timestamp": "2026-02-08T16:19:00.768145Z" }
e29d72
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 153, "completion_tokens": 2692 }, "timestamp": "2026-02-17T02:00:47.231Z", "answer": 82856 }, ...
1
[ { "lemma": "COUNT_PRIMES", "status": "ok_later" }, { "lemma": "K14", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" }, { "lemma": "MIN_PRIME_FACTOR", "status": "ok" }, { "lemma": "MOD_MUL", "status": "no" }, { "lemma": "V1", "status...
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
1af727
comb_bell_compute_v1_677425708_2813
Let $N = 2054$. Let $n$ be the number of integers $j$ with $0 \le j \le N$ such that $$\binom{2054}{j} \equiv 1 \pmod{2}.$$ Let $Q$ be the $n$th Bell number, that is, the number of ways to partition a set of $n$ elements into nonempty subsets. Compute $Q$.
4,140
graphs = [ Graph( let={ "_n": Const(2054), "n": CountOverSet(set=SolutionsSet(var=Var("j"), condition=And(Geq(Var("j"), Const(0)), Leq(Var("j"), Ref("_n")), Eq(Mod(value=Binom(n=Const(2054), k=Var("j")), modulus=Const(2)), Const(1))), domain='nonnegative_integers')), "res...
COMB
null
COMPUTE
sympy
V8
[ "V8" ]
86348e
comb_bell_compute_v1
null
8
0
[ "V8" ]
1
0.004
2026-02-08T05:17:25.636832Z
{ "verified": true, "answer": 4140, "timestamp": "2026-02-08T05:17:25.640530Z" }
74b03c
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 204, "completion_tokens": 734 }, "timestamp": "2026-02-11T23:38:55.162Z", "answer": 4140 }, { "id...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MOD_SUB", "status": "no" }, { "lemma": "V8", "status": "ok" }, { "lemma": "VAL_SUM_EQ", "status": "no" } ]
{ "lo": -1.7, "mid": 0.37, "hi": 2.2 }
e85c4a
modular_sum_quadratic_residues_v1_1439011603_2538
Let $p$ be the smallest prime divisor of $77837$. Define $r = \frac{p(p-1)}{4}$. Let $Q$ be the remainder when $44121 \cdot r$ is divided by $86701$. Find the value of $Q$.
30,747
graphs = [ Graph( let={ "_n": Const(4), "p": MinOverSet(set=SolutionsSet(var=Var("d"), condition=And(Geq(Var("d"), Const(2)), Divides(divisor=Var("d"), dividend=Const(77837))))), "result": Div(Mul(Ref("p"), Sub(Ref("p"), Const(1))), Ref("_n")), "Q": Mod(value=...
NT
null
SUM
sympy
MIN_PRIME_FACTOR
[ "MIN_PRIME_FACTOR" ]
bc3776
modular_sum_quadratic_residues_v1
null
3
0
[ "MIN_PRIME_FACTOR" ]
1
0.001
2026-02-08T16:51:15.291955Z
{ "verified": true, "answer": 30747, "timestamp": "2026-02-08T16:51:15.293388Z" }
a63d72
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": 2399 }, "timestamp": "2026-02-17T13:35:51.629Z", "answer": 30747 }, ...
1
[ { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "L3c", "status": "no" }, { "lemma": "MIN_PRIME_FACTOR", "status": "ok" }, { "lemma": "MOD_ADD", "status": "no" }, { "lemma": "V8", "status": "no" }...
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
27f30c
nt_count_coprime_v1_677425708_443
Let $n$ be a positive integer such that $1 \leq n \leq 20160$ and $\gcd(n, 19) = 1$. Let $A$ be the number of such integers $n$. Compute the value of $$ A + \phi(|A| + 1) + \tau(|A| + 1), $$ where $\phi$ denotes Euler's totient function and $\tau(k)$ denotes the number of positive divisors of $k$.
26,717
graphs = [ Graph( let={ "upper": Const(20160), "k": Const(19), "result": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), Ref("upper")), Eq(GCD(a=Var("n"), b=Ref("k")), Const(1))))), "Q": Sum(Ref("result"), Euler...
NT
null
COUNT
sympy
ONE_PHI_1
[ "ONE_PHI_1" ]
f6b5a5
nt_count_coprime_v1
null
4
0
[ "ONE_PHI_1" ]
1
1.546
2026-02-08T03:33:00.696057Z
{ "verified": true, "answer": 26717, "timestamp": "2026-02-08T03:33:02.242447Z" }
9d9ce0
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 216, "completion_tokens": 1244 }, "timestamp": "2026-02-08T20:34:56.095Z", "answer": 26717 }, { "...
1
[ { "lemma": "DS2", "status": "no" }, { "lemma": "K14", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MAX_VAL", "status": "no" }, { "lemma": "ONE_PHI_1", "status": "ok" } ]
{ "lo": -6.51, "mid": -0.31, "hi": 5.43 }
e58c9d
nt_max_prime_below_v1_1456120455_14
Let $c$ be the number of positive integers $p$ for which there exists a positive integer $q$ such that $pq = 72$, $\gcd(p, q) = 1$, and $p < q$. Let $R$ be the largest prime number $n$ such that $c \leq n \leq 46225$. Compute the remainder when $44121 \cdot R$ is divided by $95734$.
94,299
graphs = [ Graph( let={ "upper": Const(46225), "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
0.98
2026-02-08T02:48:33.214207Z
{ "verified": true, "answer": 94299, "timestamp": "2026-02-08T02:48:34.193865Z" }
c13b5d
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 204, "completion_tokens": 32768 }, "timestamp": "2026-02-23T17:42:53.657Z", "answer": 94299 }, { ...
1
[ { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "L3b", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" }, { "lemma": "MOD_SUB", ...
{ "lo": 1.88, "mid": 3.52, "hi": 5.13 }
2fb64d
comb_binomial_compute_v1_1520064083_2233
Let $n$ be the largest prime number satisfying $2 \leq n \leq 15$. Let $k$ be the largest prime number satisfying $2 \leq k \leq 6$. Compute the value of $\binom{n}{k}$.
1,287
graphs = [ Graph( let={ "_m": Const(6), "_n": Const(2), "n": MaxOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(2)), Leq(Var("n"), Const(15)), IsPrime(Var("n"))))), "k": MaxOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("...
NT
null
COMPUTE
sympy
MAX_PRIME_BELOW
[ "MAX_PRIME_BELOW" ]
dc3ad3
comb_binomial_compute_v1
null
3
0
[ "MAX_PRIME_BELOW" ]
1
0.004
2026-02-08T04:35:26.471050Z
{ "verified": true, "answer": 1287, "timestamp": "2026-02-08T04:35:26.474991Z" }
caabe6
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 167, "completion_tokens": 461 }, "timestamp": "2026-02-10T17:09:05.161Z", "answer": 1287 }, { "i...
2
[ { "lemma": "DS2", "status": "no" }, { "lemma": "K13", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok" }, { "lemma": "MOD_MUL", "status": "no" }, { "lemma": "V5", "status": "no" } ]
{ "lo": -9.14, "mid": -6.05, "hi": -3.73 }
dc8b9d
comb_count_permutations_fixed_v1_168721529_2034
Let $N$ be the smallest positive integer $n$ such that $7^{140}$ divides $n!$. Let $d_{\min}$ be the smallest integer $d \geq 2$ that divides $N$. Define $n = d_{\min}$. Compute the value of $$ \binom{n}{3} \cdot !(n - 3), $$ where $!k$ denotes the number of derangements of $k$ elements. Let $m = 44121$. Find the remai...
8,921
graphs = [ Graph( let={ "_m": Const(44121), "_n": Const(2), "n": MinOverSet(set=SolutionsSet(var=Var("d"), condition=And(Geq(Var("d"), Ref("_n")), Divides(divisor=Var("d"), dividend=MinOverSet(set=SolutionsSet(var=Var("n"), condition=Geq(MaxKDivides(target=Factorial(Var("...
NT
COMB
COUNT
sympy
V5
[ "V5/MIN_PRIME_FACTOR" ]
da0dce
comb_count_permutations_fixed_v1
null
7
0
[ "MIN_PRIME_FACTOR", "V5" ]
2
0.004
2026-02-08T14:03:35.762408Z
{ "verified": true, "answer": 8921, "timestamp": "2026-02-08T14:03:35.766611Z" }
bc6904
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 231, "completion_tokens": 1543 }, "timestamp": "2026-02-10T01:09:24.026Z", "answer": 8921 } ]
2
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "L3b", "status": "no" }, { "lemma": "MIN_PRIME_FACTOR", "status": "ok_later" ...
{ "lo": -10, "mid": -1.96, "hi": 6.09 }
5a5fbb
algebra_poly_eval_v1_601307018_1004
Let $n$ be the number of ordered pairs $(a, b)$ of positive integers with $$1 \le a \le \left|\left\{(a1, b1) : 1 \le a1 \le 40,\ 1 \le b1 \le 40,\ 68a1^{3}b1 + 17b1^{4} + \min\{ |x - y| : x > 0,\ y > 0,\ xy = 21115 \} \cdot a1^{2} b1^{\left|\{ p : p > 0,\ \exists q \in \mathbb{Z} \text{ with } pq = 72,\ \gcd(p, q) = 1...
23,872
graphs = [ Graph( let={ "_d": Const(4), "_c": Const(2), "_m": Const(3), "_n": Const(3), "n": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var("a"), Var("b")]), condition=And(Geq(Var("a"), Const(1)), Leq(Var("a"), CountOverSet(set=SolutionsSet(...
NT
null
COMPUTE
sympy
COPRIME_PAIRS
[ "COPRIME_PAIRS/POLY4_COUNT/QF_PSD_COUNT", "B3_DIFF/POLY4_COUNT/QF_PSD_COUNT" ]
e9a352
algebra_poly_eval_v1
null
8
0
[ "B3_DIFF", "COPRIME_PAIRS", "POLY4_COUNT", "QF_PSD_COUNT" ]
4
0.048
2026-03-10T01:34:43.584379Z
{ "verified": true, "answer": 23872, "timestamp": "2026-03-10T01:34:43.632752Z" }
4612f7
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 389, "completion_tokens": 3363 }, "timestamp": "2026-04-19T00:45:22.442Z", "answer": 23872 }, { ...
1
[ { "lemma": "B3_DIFF", "status": "ok" }, { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "K18", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MAX_VAL", "status": "no" }, { "lemma": "MOD_ADD", "status": "no" }, { ...
{ "lo": -1.06, "mid": 2.32, "hi": 5.24 }
4233e4
antilemma_k2_v1_1915831931_3349
Let $m = 17$. Define $n$ to be $$ \sum_{k=1}^{m} \phi(k) \left\lfloor \frac{17}{k} \right\rfloor. $$ Compute $$ \sum_{k=1}^{n} \phi(k) \left\lfloor \frac{1 + 2 + \cdots + 17}{k} \right\rfloor. $$
11,781
graphs = [ Graph( let={ "_m": Const(17), "_n": Summation(var="k", start=Const(1), end=Ref("_m"), expr=Mul(EulerPhi(n=Var("k")), Floor(Div(Const(17), Var("k"))))), "x": Summation(var="k1", start=Const(1), end=Ref("_n"), expr=Mul(EulerPhi(n=Var("k1")), Floor(Div(Summation(v...
NT
COMB
COMPUTE
sympy
K2
[ "K2/SUM_ARITHMETIC/K2", "K2" ]
85b706
antilemma_k2_v1
null
4
0
[ "K2", "SUM_ARITHMETIC" ]
2
0.002
2026-02-08T17:34:31.753626Z
{ "verified": true, "answer": 11781, "timestamp": "2026-02-08T17:34:31.755645Z" }
599f8b
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 132, "completion_tokens": 1292 }, "timestamp": "2026-02-18T04:37:34.649Z", "answer": 11781 }, ...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "K2", "status": "ok" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, { "lemma": "SUM_ARITHMETIC", "status": "ok_later" }, ...
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
b63672_l
nt_min_coprime_above_v1_1116507919_202
Let $\text{modulus}$ be the number of integers $t$ such that $18 \leq t \leq 334$ and $t = 8a + 10b$ for some integers $a$ and $b$ with $1 \leq a \leq 8$ and $1 \leq b \leq 27$. Compute the smallest integer $n$ such that $73984 < n \leq 74141$ and $\gcd(n, \text{modulus}) = 1$.
73,987
NT
null
EXTREMUM
sympy
LIN_FORM
[ "LIN_FORM" ]
7b2633
nt_min_coprime_above_v1
null
5
0
[ "LIN_FORM" ]
1
0.025
2026-02-08T02:27:52.498778Z
{ "verified": false, "answer": 73985, "timestamp": "2026-02-08T02:27:52.523602Z" }
1be7df
b63672
legacy_text
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 224, "completion_tokens": 32768 }, "timestamp": "2026-02-23T13:51:18.546Z", "answer": null }, { ...
1
[ { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "K14", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "MOD_SUB", "status": "no" }, { "lemma": "V5", "status": "no" } ]
{ "lo": 1.42, "mid": 2.89, "hi": 4.27 }
e74120
sequence_lucas_compute_v1_784195855_5286
Let $n$ be the smallest integer $d \geq 2$ that divides $6982823$. Compute the $n$th Lucas number.
9,349
graphs = [ Graph( let={ "_n": Const(2), "n": MinOverSet(set=SolutionsSet(var=Var("d"), condition=And(Geq(Var("d"), Ref("_n")), Divides(divisor=Var("d"), dividend=Const(6982823))))), "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
4
0
[ "MIN_PRIME_FACTOR" ]
1
0.001
2026-02-08T07:49:09.404500Z
{ "verified": true, "answer": 9349, "timestamp": "2026-02-08T07:49:09.405131Z" }
33b71c
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": 1621 }, "timestamp": "2026-02-13T12:31:04.213Z", "answer": 9349 }, { ...
1
[ { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "K13", "status": "no" }, { "lemma": "MIN_PRIME_FACTOR", "status": "ok" }, { "lemma": "MOD_ADD", "status": "no" }, { "lemma": "V1", "status": "no" }, { "lemma": "V8_SUM", "status": "no" } ]
{ "lo": -3.51, "mid": 1.33, "hi": 6.56 }
72cbad
modular_min_linear_v1_1520064083_6423
Let $ S $ be the set of all integers $ t $ such that there exist integers $ a $ and $ b $ satisfying $ 1 \leq a \leq 283 $, $ 1 \leq b \leq 859 $, $ 9 \leq t \leq 3699 $, and $ t = 7a + 2b $. Let $ b $ be the number of elements in $ S $. Let $ a = 16435 $ and $ m = 38593 $. Define $ x $ to be the smallest positive inte...
49,458
graphs = [ Graph( let={ "_n": Const(57140), "a": Const(16435), "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'), rig...
NT
null
EXTREMUM
sympy
LIN_FORM
[ "LIN_FORM" ]
7b2633
modular_min_linear_v1
null
6
0
[ "LIN_FORM" ]
1
1.522
2026-02-08T08:03:44.684938Z
{ "verified": true, "answer": 49458, "timestamp": "2026-02-08T08:03:46.207370Z" }
59574c
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 217, "completion_tokens": 5506 }, "timestamp": "2026-02-13T14:23:58.280Z", "answer": 49458 }, ...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "V5", "status": "no" }, { "lemma": "V7", "status": "no" } ]
{ "lo": -3.44, "mid": 1.34, "hi": 6.56 }
b893fd_n
alg_telescope_v1_1218484723_2844
A robot computes a cumulative score by summing the values $4k^3 + 6k^2 + 4k + 1$ for $k = 0$ to $1549$, then takes the remainder modulo the sum of $4k_1 + 199$ for $k_1 = 0$ to $17$. Let $M$ be this final value. The robot then multiplies $M$ by $44121$ and computes the remainder when divided by $50306$. What is this fi...
37,450
ALG
null
COMPUTE
sympy
SUM_AP
[ "SUM_AP" ]
ff6f57
alg_telescope_v1
null
4
null
[ "SUM_AP" ]
1
0.124
2026-02-25T04:34:51.445663Z
null
ecd788
b893fd
narrative
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 221, "completion_tokens": 5244 }, "timestamp": "2026-03-30T19:08:36.909Z", "answer": 37450 }, { "...
1
[ { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" }, { "lemma": "SUM_AP", "status": "ok" } ]
{ "lo": -5.37, "mid": 0.23, "hi": 5.22 }
679b8b
antilemma_cartesian_v1_1125832087_2416
Let $S$ be the set of all ordered pairs $(a,b)$ such that $a$ is an integer with $1 \leq a \leq 18$ and $b$ is an integer with $1 \leq b \leq 21$. Compute the number of elements in $S$.
378
graphs = [ Graph( let={ "x": CountOverSet(set=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(18)), right=IntegerRange(start=Const(1), end=Const(21)))), }, goal=Ref("x"), ) ]
COMB
GEOM
COMPUTE
sympy
COUNT_CARTESIAN
[ "COUNT_CARTESIAN" ]
409d7d
antilemma_cartesian_v1
null
2
0
[ "COUNT_CARTESIAN" ]
1
0.002
2026-02-08T04:35:40.956564Z
{ "verified": true, "answer": 378, "timestamp": "2026-02-08T04:35:40.958575Z" }
4f245f
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 175, "completion_tokens": 231 }, "timestamp": "2026-02-24T01:17:49.147Z", "answer": 378 }, { "id"...
2
[ { "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", "status": ...
{ "lo": -10, "mid": -7.79, "hi": -5.57 }
0b6e31
sequence_count_fib_divisible_v1_1470522791_1191
Let $T$ be the set of all integers $t$ with $18 \leq t \leq 1834$ for which there exist positive integers $a \leq 123$ and $b \leq 85$ such that $t = 8a + 10b$. Let $u$ be the number of elements in $T$. Let $d$ be the largest prime number at most 16. Determine the number of positive integers $n \leq u$ such that $d$ di...
128
graphs = [ Graph( let={ "_n": Const(16), "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=123)), Geq(left...
NT
null
COUNT
sympy
MAX_PRIME_BELOW
[ "MAX_PRIME_BELOW", "LIN_FORM" ]
d530e2
sequence_count_fib_divisible_v1
null
6
0
[ "LIN_FORM", "MAX_PRIME_BELOW" ]
2
0.042
2026-02-08T13:29:40.745269Z
{ "verified": true, "answer": 128, "timestamp": "2026-02-08T13:29:40.787758Z" }
53348b
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 159, "completion_tokens": 4941 }, "timestamp": "2026-02-15T16:59:12.535Z", "answer": 128 }, { ...
1
[ { "lemma": "K16", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok" }, { "lemma": "MOD_ADD", "status": "no" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, { "lemma": "V1", "status": "no" }, ...
{ "lo": -5.49, "mid": 0.22, "hi": 6.51 }
693f13
modular_modexp_compute_v1_677425708_2065
Let $a = 47$. Let $e$ be the number of positive integers $j$ such that $1 \leq j \leq 2304$ and $j^2 \leq 5308416$. Let $m = 11449$. Define $\text{result}$ to be $a^e \bmod m$, that is, the remainder when $a^e$ is divided by $m$. Compute $\text{result}$.
7,148
graphs = [ Graph( let={ "a": Const(47), "e": CountOverSet(set=SolutionsSet(var=Var("j"), condition=And(Geq(Var("j"), Const(1)), Leq(Var("j"), Const(2304)), Leq(Pow(Var("j"), Const(2)), Const(5308416))), domain='positive_integers')), "m": Const(11449), "result"...
NT
null
COMPUTE
sympy
C3
[ "C3" ]
8a214c
modular_modexp_compute_v1
null
4
0
[ "C3" ]
1
0.001
2026-02-08T04:44:15.165219Z
{ "verified": true, "answer": 7148, "timestamp": "2026-02-08T04:44:15.166059Z" }
29d03f
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 207, "completion_tokens": 5238 }, "timestamp": "2026-02-11T07:25:44.164Z", "answer": 7148 }, { "...
1
[ { "lemma": "C3", "status": "ok" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "K18", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, { "lemma": "MOD_SUB", "status": "no" ...
{ "lo": -3.42, "mid": 1.86, "hi": 7.55 }
94fb44
modular_mod_compute_v1_153355830_1705
Let $a = -61009$ and $m = 69169$. Compute $r$, the remainder when $a$ is divided by $m$. Let $T$ be the set of all integers $t$ such that $12 \leq t \leq 334$ and there exist integers $a$ and $b$ with $1 \leq a \leq 25$, $1 \leq b \leq 78$, and $t = 7a + 2b + 3$. Find the remainder when $$ (r \bmod 307) + 5003 \cdot (r...
12,794
graphs = [ Graph( let={ "a": Const(-61009), "m": Const(69169), "result": Mod(value=Ref("a"), modulus=Ref("m")), "Q": Mod(value=Sum(Mod(value=Ref("result"), modulus=Const(307)), Mul(Const(5003), Mod(value=Ref("result"), modulus=CountOverSet(set=SolutionsSet(var...
NT
null
COMPUTE
sympy
LIN_FORM
[ "LIN_FORM" ]
d6c893
modular_mod_compute_v1
two_moduli
4
0
[ "LIN_FORM" ]
1
0.001
2026-02-08T06:34:07.208550Z
{ "verified": true, "answer": 12794, "timestamp": "2026-02-08T06:34:07.209833Z" }
180bb6
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 187, "completion_tokens": 3049 }, "timestamp": "2026-02-13T01:48:37.436Z", "answer": 12794 }, ...
1
[ { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "DS2", "status": "no" }, { "lemma": "K18", "status": "no" }, { "lemma": "L3b", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" } ...
{ "lo": -3.44, "mid": 1.34, "hi": 6.56 }
10d441
modular_min_linear_v1_124444284_1931
Let $A$ be the set of all integers $n$ with $1 \leq n \leq 31239$ such that $n \equiv \left\lfloor \frac{n}{2} \right\rfloor \pmod{11}$. Let $a = |A|$. Let $B$ be the set of all integers $t$ with $10 \leq t \leq 2300$ such that there exist positive integers $a \leq 74$ and $b \leq 594$ satisfying $t = 7a + 3b$. Let $b ...
3,275
graphs = [ Graph( let={ "_n": Const(63454), "a": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), Const(31239)), Congruent(a=Var(name='n'), b=Floor(arg=Div(left=Var(name='n'), right=Const(value=2))), modulus=Const(value=11))))), ...
NT
null
EXTREMUM
sympy
LIN_FORM
[ "LIN_FORM", "L3C" ]
ecf77f
modular_min_linear_v1
null
7
0
[ "L3C", "LIN_FORM" ]
2
0.669
2026-02-08T04:12:51.080237Z
{ "verified": true, "answer": 3275, "timestamp": "2026-02-08T04:12:51.749155Z" }
0cd854
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 232, "completion_tokens": 5877 }, "timestamp": "2026-02-11T22:41:24.615Z", "answer": 3275 }, {...
1
[ { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "L3C", "status": "ok" }, { "lemma": "L3c", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { ...
{ "lo": -3.44, "mid": 1.34, "hi": 6.56 }
598a13
modular_min_modexp_v1_1125832087_1083
Let $x$ be the smallest positive integer such that $1 \leq x \leq 52$ and $2^x \equiv 7 \pmod{157}$. Let $r$ be the remainder when $x$ is divided by 11. Compute the $r$-th Bell number.
1
graphs = [ Graph( let={ "a": Const(2), "b": Const(7), "m": Const(157), "upper": Const(52), "result": MinOverSet(set=SolutionsSet(var=Var("x"), condition=And(Geq(Var("x"), Const(1)), Leq(Var("x"), Ref("upper")), Eq(ModExp(base=Ref("a"), exp=Var("x")...
NT
COMB
EXTREMUM
sympy
MIN_PRIME_FACTOR
[ "MIN_PRIME_FACTOR" ]
58d7e9
modular_min_modexp_v1
bell_mod
6
0
[ "MIN_PRIME_FACTOR" ]
1
1.749
2026-02-08T03:30:49.187233Z
{ "verified": true, "answer": 1, "timestamp": "2026-02-08T03:30:50.936258Z" }
58728f
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 177, "completion_tokens": 627 }, "timestamp": "2026-02-10T14:51:08.687Z", "answer": 1 }, { "id": ...
1
[ { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "K13", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "MIN_PRIME_FACTOR", "status": "ok" }, { "lemma": "V5", "status": "no" ...
{ "lo": -3.46, "mid": 1.03, "hi": 5.49 }
61535d
geo_count_lattice_triangle_v1_1978505735_2877
Let $A$ be the area of the triangle with vertices at $(0,0)$, $(180,66)$, and $(111,128)$, multiplied by $2$. Let $B$ be the sum of the greatest common divisors of the absolute differences in coordinates along each side of the triangle: - $\gcd(|180 - 0|, |66 - 0|)$, - $\gcd(|111 - 180|, |128 - 66|)$, - $\gcd(|0 - 111|...
78,842
graphs = [ Graph( let={ "area_2x": Abs(arg=Sum(Mul(Const(value=180), Const(value=128)), Mul(Const(value=111), Sub(left=Const(value=0), right=Const(value=66))))), "boundary": Sum(GCD(a=Abs(arg=Const(value=180)), b=Abs(arg=Const(value=66))), GCD(a=Abs(arg=Sub(left=Const(value=111), rig...
ALG
NT
COUNT
sympy
[]
geo_count_lattice_triangle_v1
null
5
0
null
null
0.003
2026-02-08T17:13:54.752218Z
{ "verified": true, "answer": 78842, "timestamp": "2026-02-08T17:13:54.755499Z" }
a07bf7
CC BY 4.0
[ { "id": 8, "model": "mathstral", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 241, "completion_tokens": 598 }, "timestamp": "2026-02-16T09:10:58.753Z", "answer": null }, { "id": 11,...
1
[]
{ "lo": -8.26, "mid": -5.01, "hi": -1.76 }
b953ec
sequence_count_fib_divisible_v1_1918700295_1964
Let $S$ be the set of positive integers $n$ such that $1 \le n \le 6160$ and $n \equiv \left\lfloor \frac{n}{2} \right\rfloor \pmod{11}$. Let $N = |S|$. Compute the number of positive integers $n$ such that $1 \le n \le N$ and the $n$th Fibonacci number is divisible by 11.
56
graphs = [ Graph( let={ "_n": Const(6160), "upper": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), Ref("_n")), Congruent(a=Var(name='n'), b=Floor(arg=Div(left=Var(name='n'), right=Const(value=2))), modulus=Const(value=11))))), ...
NT
null
COUNT
sympy
L3C
[ "L3C" ]
73f8b0
sequence_count_fib_divisible_v1
null
6
0
[ "L3C" ]
1
0.085
2026-02-08T06:12:00.697666Z
{ "verified": true, "answer": 56, "timestamp": "2026-02-08T06:12:00.782944Z" }
90d469
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 140, "completion_tokens": 1183 }, "timestamp": "2026-02-13T11:27:19.652Z", "answer": 55 }, ...
1
[ { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "L3C", "status": "ok" }, { "lemma": "MOD_MUL", "status": "no" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" }, { "lemma": "V8", "status": "no"...
{ "lo": -3.44, "mid": 1.34, "hi": 6.56 }
5aaee8
antilemma_k3_v1_784195855_9458
Let $x = \sum_{d \mid 50724} \phi(d)$, where $\phi$ denotes Euler's totient function. Compute the remainder when $28226 \cdot x$ is divided by $64983$.
30,168
graphs = [ Graph( let={ "x": SumOverDivisors(n=Const(value=50724), var='d', expr=EulerPhi(n=Var(name='d'))), "_c": Const(28226), "Q": Mod(value=Mul(Ref("_c"), Ref("x")), modulus=Const(64983)), }, goal=Ref("Q"), ) ]
NT
COMB
COMPUTE
sympy
K3
[ "K3" ]
54c41e
antilemma_k3_v1
null
3
0
[ "K3" ]
1
0.001
2026-02-08T16:50:37.472333Z
{ "verified": true, "answer": 30168, "timestamp": "2026-02-08T16:50:37.473006Z" }
de661b
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 93, "completion_tokens": 1395 }, "timestamp": "2026-02-17T13:44:02.425Z", "answer": 30168 }, {...
1
[ { "lemma": "K14", "status": "no" }, { "lemma": "K18", "status": "no" }, { "lemma": "K3", "status": "ok" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" }, { "lemma": "MOD_MUL", "status": "no" } ]
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
5ede2d
comb_bell_compute_v1_655260480_4830
Let $n$ be the sum of all real solutions $x$ to the equation $x^2 - 8x - 9588 = 0$. Compute the Bell number $B_n$, which counts the number of partitions of a set of $n$ elements.
4,140
graphs = [ Graph( let={ "_n": Const(2), "n": SumOverSet(set=SolutionsSet(var=Var("x"), condition=Eq(Sum(Pow(Var("x"), Ref("_n")), Mul(Const(-8), Var("x")), Const(-9588)), Const(0)))), "result": Bell(Ref("n")), }, goal=Ref("result"), ) ]
COMB
null
COMPUTE
sympy
K2
[ "VIETA_SUM" ]
b33a7a
comb_bell_compute_v1
null
3
0
[ "K2", "VIETA_SUM" ]
2
0.008
2026-02-08T18:08:53.591186Z
{ "verified": true, "answer": 4140, "timestamp": "2026-02-08T18:08:53.599452Z" }
5c512f
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": 678 }, "timestamp": "2026-02-18T14:53:44.069Z", "answer": 4140 }, { ...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "V8", "status": "no" }, { "lemma": "V8_SUM", "status": "no" }, { "lemma": "VIETA_SUM", "status": "o...
{ "lo": -3.12, "mid": 1.47, "hi": 6.57 }
00ec85
diophantine_product_count_v1_677425708_2816
Let $k = 180$ and $u = 47$. Compute the number of positive integers $x$ such that $1 \leq x \leq u$, $x$ divides $k$, and $\frac{k}{x} \leq u$. Find the value of this count.
12
graphs = [ Graph( let={ "k": Const(180), "upper": Const(47), "result": CountOverSet(set=SolutionsSet(var=Var("x"), condition=And(Geq(Var("x"), Const(1)), Leq(Var("x"), Ref("upper")), Divides(divisor=Var("x"), dividend=Ref("k")), Leq(Div(Ref("k"), Var("x")), Ref("upper")))...
NT
null
COUNT
sympy
LIN_FORM
[ "LIN_FORM", "B3" ]
688dbe
diophantine_product_count_v1
null
3
0
[ "B3", "LIN_FORM" ]
2
0.062
2026-02-08T05:17:26.202808Z
{ "verified": true, "answer": 12, "timestamp": "2026-02-08T05:17:26.264896Z" }
718605
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 106, "completion_tokens": 778 }, "timestamp": "2026-02-12T06:31:40.406Z", "answer": 12 }, { ...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "K18", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "MOD_SUB", "status": "no" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8_SUM...
{ "lo": -5.55, "mid": -3.01, "hi": 0.32 }
0e31b5
antilemma_k3_v1_1520064083_8723
Let $n = 32830$. Compute the remainder when $44121$ times the sum $\sum_{d \mid n} \phi(d)$ is divided by $79654$, where $\phi$ denotes Euler's totient function.
64,094
graphs = [ Graph( let={ "_n": Const(32830), "x": SumOverDivisors(n=Ref(name='_n'), var='d', expr=EulerPhi(n=Var(name='d'))), "Q": Mod(value=Mul(Const(44121), Ref("x")), modulus=Const(79654)), }, goal=Ref("Q"), ) ]
NT
COMB
COMPUTE
sympy
K3
[ "K3" ]
54c41e
antilemma_k3_v1
null
4
0
[ "K3" ]
1
0.001
2026-02-08T10:20:43.902473Z
{ "verified": true, "answer": 64094, "timestamp": "2026-02-08T10:20:43.903444Z" }
2ede5e
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 95, "completion_tokens": 1081 }, "timestamp": "2026-02-14T07:09:05.883Z", "answer": 64094 }, {...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "K14", "status": "no" }, { "lemma": "K15", "status": "no" }, { "lemma": "K3", "status": "ok" }, { "lemma": "K5", "status": "no" }, { "lemma": "V7", "status": "no" } ]
{ "lo": -5.14, "mid": 0.32, "hi": 6.51 }
01677e
antilemma_cartesian_v1_124444284_1582
Compute the remainder when $48512$ times the number of ordered pairs $(i,j)$ with $1 \leq i \leq 34$ and $1 \leq j \leq 39$ is divided by $78933$.
75,450
graphs = [ Graph( let={ "x": CountOverSet(set=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(34)), right=IntegerRange(start=Const(1), end=Const(39)))), "Q": Mod(value=Mul(Const(48512), Ref("x")), modulus=Const(78933)), }, goal=Ref("Q"), ) ]
COMB
GEOM
COMPUTE
sympy
COUNT_CARTESIAN
[ "COUNT_CARTESIAN" ]
409d7d
antilemma_cartesian_v1
null
2
0
[ "COUNT_CARTESIAN" ]
1
0.001
2026-02-08T04:01:43.978596Z
{ "verified": true, "answer": 75450, "timestamp": "2026-02-08T04:01:43.979473Z" }
f22377
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 161, "completion_tokens": 984 }, "timestamp": "2026-02-11T15:47:49.685Z", "answer": 75450 }, { "i...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_CARTESIAN", "status": "ok" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8_SUM", "status":...
{ "lo": -2.46, "mid": 0.47, "hi": 3.59 }
92e60e
nt_count_coprime_v1_2051736721_162
Let $n = 5$ and let $k$ be the value of $$\sum_{k=1}^{5} \phi(k) \left\lfloor \frac{5}{k} \right\rfloor,$$ where $\phi$ denotes Euler's totient function. Let $S$ be the set of all positive integers $n$ such that $1 \leq n \leq 58564$ and $\gcd(n, k) = 1$. Compute the number of elements in $S$.
31,235
graphs = [ Graph( let={ "_n": Const(5), "upper": Const(58564), "k": Summation(var="k1", start=Const(1), end=Ref("_n"), expr=Mul(EulerPhi(n=Var("k1")), Floor(Div(Const(5), Var("k1"))))), "result": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Va...
NT
null
COUNT
sympy
K2
[ "K2" ]
6897ab
nt_count_coprime_v1
null
5
0
[ "K2" ]
1
7.965
2026-02-08T15:15:47.665392Z
{ "verified": true, "answer": 31235, "timestamp": "2026-02-08T15:15:55.630867Z" }
a72d1c
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": 1535 }, "timestamp": "2026-02-16T02:30:47.405Z", "answer": 31235 }, ...
1
[ { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "K15", "status": "no" }, { "lemma": "K2", "status": "ok" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" }, { "lemma": "MOD_ADD", "status": "no" ...
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
849007
diophantine_fbi2_min_v1_397696148_2635
Let $k = 12$ and $\text{upper} = 22$. Define $\text{result}$ to be the smallest integer $d$ such that $2 \le d \le \text{upper}$, $d$ divides $k$, and $\frac{k}{d}$ is at least the number of nonnegative integers $j$ with $0 \le j \le 4128$ for which $\binom{4128}{j} \equiv 1 \pmod{2}$. Compute $\text{result}$.
2
graphs = [ Graph( let={ "_n": Const(2), "k": Const(12), "upper": Const(22), "result": MinOverSet(set=SolutionsSet(var=Var("d"), condition=And(Geq(Var("d"), Const(2)), Leq(Var("d"), Ref("upper")), Divides(divisor=Var("d"), dividend=Ref("k")), Geq(Div(Ref("k"), ...
NT
null
EXTREMUM
sympy
V8
[ "V8" ]
86348e
diophantine_fbi2_min_v1
null
6
0
[ "V8" ]
1
0.007
2026-02-08T13:27:06.956038Z
{ "verified": true, "answer": 2, "timestamp": "2026-02-08T13:27:06.963488Z" }
c41826
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": 1437 }, "timestamp": "2026-02-15T15:30:17.367Z", "answer": 2 }, { ...
1
[ { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "K14", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "V8", "status": "ok" }, { "lemma": "VAL_SUM_EQ", "status": "no" } ]
{ "lo": -8.26, "mid": -5.01, "hi": -1.76 }
ef03ec
algebra_vieta_sum_v1_1742523217_5160
Let $f(x) = -x^3 - 7x^2 - 4x + c$, where $c$ is the number of ordered pairs $(x_1, x_2)$ of positive odd integers such that $x_1 + x_2 = 24$. Find the product of all real roots of the equation $f(x) = 0$.
12
graphs = [ Graph( let={ "_n": Const(3), "result": ProductOverSet(set=SolutionsSet(var=Var(name='x'), condition=Eq(left=Sum(Mul(Const(value=-1), Pow(base=Var(name='x'), exp=Ref(name='_n'))), Mul(Const(value=-7), Pow(base=Var(name='x'), exp=Const(value=2))), Mul(Const(value=-4), Var(na...
NT
null
COMPUTE
sympy
MAX_PRIME_BELOW
[ "COMB1" ]
567f58
algebra_vieta_sum_v1
null
6
0
[ "COMB1", "MAX_PRIME_BELOW" ]
2
0.024
2026-02-08T10:50:31.071306Z
{ "verified": true, "answer": 12, "timestamp": "2026-02-08T10:50:31.095180Z" }
cdd4df
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 125, "completion_tokens": 764 }, "timestamp": "2026-02-14T08:58:14.559Z", "answer": 12 }, { ...
1
[ { "lemma": "COMB1", "status": "ok" }, { "lemma": "DS2", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "V8", "status": "no" } ]
{ "lo": -5.49, "mid": 0.22, "hi": 6.51 }
a6e23e
alg_linear_system_2x2_v1_1218484723_5100
Let $\det = 16 \cdot 4 - 15 \cdot 16$ and $M = 483456 \cdot 4 - 452151 \cdot 16$. Let $R = \min\{ 29b^2 + 29a^2 - 42ab \mid 1 \leq a \leq 5,\ 1 \leq b \leq 5\} \cdot 452151 - 15 \cdot 483456$. Compute $\frac{M}{\det} + \frac{R}{\det}$.
30,216
graphs = [ Graph( let={ "_n": Const(16), "num_x": Sub(Mul(Const(483456), Const(4)), Mul(Const(452151), Const(16))), "num_y": Sub(Mul(MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var("a"), Var("b")]), condition=And(Geq(Var("a"), Const(1)), Leq(Var("a"), C...
ALG
null
COMPUTE
sympy
QF_PSD_MIN
[ "QF_PSD_MIN" ]
c66ce3
alg_linear_system_2x2_v1
null
3
0
[ "QF_PSD_MIN" ]
1
0.002
2026-02-25T06:43:54.084754Z
{ "verified": true, "answer": 30216, "timestamp": "2026-02-25T06:43:54.087181Z" }
156611
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 237, "completion_tokens": 2055 }, "timestamp": "2026-03-29T19:26:03.791Z", "answer": 30216 }, { "...
1
[ { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" }, { "lemma": "QF_PSD_MIN", "status": "ok" } ]
{ "lo": -3.33, "mid": 1.03, "hi": 5.26 }
bbaaea
comb_count_partitions_v1_1470522791_1195
Let $n$ be the number of integers $t$ such that $15 \le t \le 138$ and there exist positive integers $a$ and $b$ with $1 \le a \le 11$, $1 \le b \le 8$, and $t = 6a + 9b$. Define $p(n)$ to be the number of integer partitions of $n$. Compute $p(n)$.
37,338
graphs = [ Graph( let={ "n": CountOverSet(set=SolutionsSet(var=Var("t"), condition=Exists(var=Var(name='a'), condition=Exists(var=Var(name='b'), condition=And(Geq(left=Var(name='a'), right=Const(value=1)), Leq(left=Var(name='a'), right=Const(value=11)), Geq(left=Var(name='b'), right=Const(value=...
COMB
null
COUNT
sympy
LIN_FORM
[ "LIN_FORM" ]
7b2633
comb_count_partitions_v1
null
5
0
[ "LIN_FORM" ]
1
0.002
2026-02-08T13:30:44.713356Z
{ "verified": true, "answer": 37338, "timestamp": "2026-02-08T13:30:44.715213Z" }
195d77
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 201, "completion_tokens": 2393 }, "timestamp": "2026-02-24T18:31:35.361Z", "answer": 37338 }, { "...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8_SUM", "status": "no" ...
{ "lo": 1.15, "mid": 4.18, "hi": 6.61 }
c11e1b
diophantine_product_count_v1_124444284_8446
Let $k = 840$. Let $S$ be the set of all ordered pairs $(x, y)$ of positive integers such that $x + y = 30$. Define $M$ to be the maximum value of $xy$ over all such pairs. Compute the number of positive integers $x \leq M$ such that $x$ divides $k$ and $\frac{k}{x} \leq M$. Let this count be $c$. Find $85849 - c$.
85,823
graphs = [ Graph( let={ "k": Const(840), "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(30)))), expr=Mul(Var("x"), Var("y")))),...
NT
null
COUNT
sympy
B1
[ "B1" ]
5b950e
diophantine_product_count_v1
null
5
0
[ "B1" ]
1
0.01
2026-02-08T09:42:40.666850Z
{ "verified": true, "answer": 85823, "timestamp": "2026-02-08T09:42:40.676562Z" }
4a11a8
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 147, "completion_tokens": 1352 }, "timestamp": "2026-02-14T05:44:28.149Z", "answer": 85823 }, ...
1
[ { "lemma": "B1", "status": "ok" }, { "lemma": "C2", "status": "no" }, { "lemma": "K14", "status": "no" }, { "lemma": "MOD_SUB", "status": "no" }, { "lemma": "V3", "status": "no" }, { "lemma": "V7", "status": "no" } ]
{ "lo": -5.14, "mid": 0.32, "hi": 6.51 }
992ec2
v1_endings_v1_1248542787_302
Let $n = 35340$ and $p = 5$. Let $n!$ denote the factorial of $n$, and let $v_p(n!)$ be the largest integer $k$ such that $p^k$ divides $n!$. Compute the remainder when $v_p(n!)$ is divided by $50761$.
8,832
graphs = [ Graph( let={ "n_val": Const(35340), "p_val": Const(5), "n_fact": Factorial(Ref("n_val")), "vp": MaxKDivides(target=Ref("n_fact"), base=Ref("p_val")), "mod_val": Const(50761), "x": Mod(value=Ref("vp"), modulus=Ref("mod_val")),...
NT
null
COMPUTE
sympy
V1
[ "V1" ]
dae96f
v1_endings_v1
null
5
null
[ "V1" ]
1
0
2026-02-08T03:03:13.265201Z
{ "verified": true, "answer": 8832, "timestamp": "2026-02-08T03:03:13.265632Z" }
2b8e26
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 185, "completion_tokens": 825 }, "timestamp": "2026-02-09T02:32:35.270Z", "answer": 8832 }, { "id...
1
[ { "lemma": "K18", "status": "no" }, { "lemma": "K5", "status": "same_pattern_wrong" }, { "lemma": "L3b", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, { "lemma": "POLY_PADIC_VAL_CONST", "st...
{ "lo": -6.51, "mid": -0.31, "hi": 5.43 }
539079
antilemma_coprime_grid_v1_168721529_861
Let $x$ be the number of ordered pairs $(i,j)$ of integers with $1 \leq i \leq 10$ and $1 \leq j \leq 130$ such that $\gcd(i,j) = \phi(1)$, where $\phi$ denotes Euler's totient function. Compute the remainder when $28343x$ is divided by $52718$.
25,500
graphs = [ Graph( let={ "x": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var("i"), Var("j")]), condition=Eq(GCD(a=Var("i"), b=Var("j")), EulerPhi(n=Const(1))), domain=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(10)), right=IntegerRange(start=Const(1), end=Const(130))))), ...
NT
COMB
COMPUTE
sympy
COUNT_COPRIME_GRID
[ "COUNT_COPRIME_GRID", "ONE_PHI_1" ]
3d404c
antilemma_coprime_grid_v1
null
3
0
[ "COUNT_COPRIME_GRID", "ONE_PHI_1" ]
2
0.002
2026-02-08T13:19:25.485479Z
{ "verified": true, "answer": 25500, "timestamp": "2026-02-08T13:19:25.487673Z" }
fea7c1
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 198, "completion_tokens": 3110 }, "timestamp": "2026-02-09T10:00:51.806Z", "answer": 25500 }, { "...
1
[ { "lemma": "COUNT_COPRIME_GRID", "status": "ok" }, { "lemma": "K14", "status": "no" }, { "lemma": "L3c", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MOD_ADD", "status": "no" }, { "lemma": "ONE_PHI_1", "status": "ok" }, ...
{ "lo": -1.99, "mid": 2.14, "hi": 6.33 }
c42be0
antilemma_sum_equals_v1_48377204_1985
Let $t$ be an integer satisfying $5 \leq t \leq 103$. Let $n$ be the number of such integers $t$ for which there exist positive integers $a$ and $b$, with $1 \leq a \leq 26$ and $1 \leq b \leq 17$, such that $t = 2a + 3b$. Compute the number of ordered pairs $(i, j)$ of integers such that $1 \leq i \leq 96$, $1 \leq j ...
96
graphs = [ Graph( let={ "_n": CountOverSet(set=SolutionsSet(var=Var("t"), condition=Exists(var=Var(name='a'), condition=Exists(var=Var(name='b'), condition=And(Geq(left=Var(name='a'), right=Const(value=1)), Leq(left=Var(name='a'), right=Const(value=26)), Geq(left=Var(name='b'), right=Const(value...
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
0.006
2026-02-08T16:32:24.453421Z
{ "verified": true, "answer": 96, "timestamp": "2026-02-08T16:32:24.459246Z" }
732569
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 169, "completion_tokens": 3729 }, "timestamp": "2026-02-17T06:25:36.438Z", "answer": 96 }, { ...
1
[ { "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_FORM", "status": "ok" }, { "lemma": ...
{ "lo": -5.64, "mid": -3.17, "hi": -0.81 }
ae7d74
antilemma_sum_equals_v1_1918700295_3978
Let $c = 160$. Consider the set of all ordered pairs $(x_1, x_2)$ of positive odd integers such that $x_1 + x_2 = c$. Let $m$ be the number of such pairs. Now consider the set of all ordered pairs $(i, j)$ of integers with $1 \leq i \leq 80$ and $1 \leq j \leq 80$ such that $i + j = m$. Let $n$ be the number of such pa...
78
graphs = [ Graph( let={ "_c": Const(160), "_m": 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
GEOM
COMPUTE
sympy
LIN_FORM
[ "COUNT_SUM_EQUALS/COUNT_SUM_EQUALS", "COMB1/COUNT_SUM_EQUALS", "COUNT_SUM_EQUALS" ]
a57484
antilemma_sum_equals_v1
null
6
0
[ "COMB1", "COUNT_SUM_EQUALS", "LIN_FORM" ]
3
0.05
2026-02-08T09:04:07.413572Z
{ "verified": true, "answer": 78, "timestamp": "2026-02-08T09:04:07.463805Z" }
6a24ca
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 313, "completion_tokens": 2620 }, "timestamp": "2026-02-24T10:27:10.683Z", "answer": 78 }, { "id"...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COMB1", "status": "ok" }, { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "ok" }, { "lemma": "V8", "status": ...
{ "lo": -2.35, "mid": 1.22, "hi": 4.84 }
943402
diophantine_fbi2_count_v1_1978505735_23
Let $k = 240$. Determine the number of positive integers $d$ such that $6 \leq d \leq 116$, $d$ divides $k$, and $6 \leq \frac{k}{d} \leq 116$. Let $r$ be this count. Compute the remainder when $44121 \cdot r$ is divided by $77965$.
51,385
graphs = [ Graph( let={ "_n": Const(44121), "k": Const(240), "result": CountOverSet(set=SolutionsSet(var=Var("d"), condition=And(Geq(Var("d"), Const(6)), Leq(Var("d"), Const(116)), Divides(divisor=Var("d"), dividend=Ref("k")), Geq(Div(Ref("k"), Var("d")), Const(6)), Leq(D...
NT
null
COUNT
sympy
MAX_PRIME_BELOW
[ "MAX_PRIME_BELOW" ]
dc3ad3
diophantine_fbi2_count_v1
null
3
0
[ "MAX_PRIME_BELOW" ]
1
0.009
2026-02-08T15:08:47.466309Z
{ "verified": true, "answer": 51385, "timestamp": "2026-02-08T15:08:47.475412Z" }
7f8725
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 192, "completion_tokens": 1631 }, "timestamp": "2026-02-10T06:44:23.548Z", "answer": 51485 }, { ...
1
[ { "lemma": "K13", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "L3b", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok" }, { "lemma": "MOD_MUL", "status": "no" } ]
{ "lo": 1.69, "mid": 5.03, "hi": 8.38 }
8cd155
comb_binomial_compute_v1_1915831931_2619
Let $n = 13$. Let $k$ be the smallest divisor of 35 that is at least 2. Compute $\binom{n}{k}$.
1,287
graphs = [ Graph( let={ "n": Const(13), "k": MinOverSet(set=SolutionsSet(var=Var("d"), condition=And(Geq(Var("d"), Const(2)), Divides(divisor=Var("d"), dividend=Const(35))))), "result": Binom(n=Ref("n"), k=Ref("k")), }, goal=Ref("result"), ) ]
NT
null
COMPUTE
sympy
MIN_PRIME_FACTOR
[ "MIN_PRIME_FACTOR" ]
bc3776
comb_binomial_compute_v1
null
2
0
[ "MIN_PRIME_FACTOR" ]
1
0.001
2026-02-08T16:59:22.161133Z
{ "verified": true, "answer": 1287, "timestamp": "2026-02-08T16:59:22.162561Z" }
df266e
CC BY 4.0
[ { "id": 8, "model": "mathstral", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 92, "completion_tokens": 610 }, "timestamp": "2026-02-16T08:54:43.474Z", "answer": 1287 }, { "id": 11, ...
2
[ { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "K14", "status": "no" }, { "lemma": "L3b", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MIN_PRIME_FACTOR", "status": "ok" }, { "lemma": "VAL_SUM_EQ", "status": ...
{ "lo": -10, "mid": -7.27, "hi": -4.54 }
b09421
antilemma_k3_v1_971394319_1090
Let $n = 57641$. Compute the sum of $\phi(d)$ over all positive divisors $d$ of $n$, where $\phi(d)$ denotes Euler's totient function.
57,641
graphs = [ Graph( let={ "x": SumOverDivisors(n=Const(value=57641), 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
0.001
2026-02-08T13:29:45.566777Z
{ "verified": true, "answer": 57641, "timestamp": "2026-02-08T13:29:45.567314Z" }
33096b
CC BY 4.0
[ { "id": 8, "model": "mathstral", "score": 1, "correct": { "strict": false, "boxed": false, "relaxed": true }, "usage": { "prompt_tokens": 102, "completion_tokens": 837 }, "timestamp": "2026-02-16T04:47:43.676Z", "answer": 30795 }, { "id": 11,...
1
[ { "lemma": "K3", "status": "ok" }, { "lemma": "K5", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "V3", "status": "no" }, { "lemma": "V8", "status": "no" } ]
{ "lo": -8.26, "mid": -5.01, "hi": -1.76 }
8d0bf7
alg_sum_powers_v1_1218484723_4527
Let $c = \left|\{ (a, b) : 1 \leq a \leq b \leq 35,\ 2a^2 - 4ab + 2b^2 = 50 \}\right|$, and let $d = \left|\{ n : 1 \leq n \leq 109260,\ c \mid F_n \}\right|$ where $F_n$ is the $n$-th Fibonacci number. Let $R = \left( \sum_{k=1}^{1388} k^2 \right) \bmod d$. Find the remainder when $83161 \cdot R$ is divided by $93616$...
10,915
graphs = [ Graph( let={ "_n": Const(1388), "result": Mod(value=Summation(var="k", start=Const(1), end=Ref("_n"), expr=Pow(Var("k"), Const(2))), modulus=CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), Const(109260)), Divides(divisor=Cou...
NT
null
COMPUTE
sympy
QF_PSD_ORBIT
[ "QF_PSD_ORBIT/COUNT_FIB_DIVISIBLE" ]
50cefd
alg_sum_powers_v1
null
5
0
[ "COUNT_FIB_DIVISIBLE", "QF_PSD_ORBIT" ]
2
0.057
2026-02-25T06:11:44.891247Z
{ "verified": true, "answer": 10915, "timestamp": "2026-02-25T06:11:44.947895Z" }
55640e
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 259, "completion_tokens": 8906 }, "timestamp": "2026-03-29T16:06:56.207Z", "answer": 57212 }, { ...
1
[ { "lemma": "COUNT_FIB_DIVISIBLE", "status": "ok_later" }, { "lemma": "K14", "status": "no" }, { "lemma": "L3b", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MAX_VAL", "status": "no" }, { "lemma": "QF_PSD_ORBIT", "status": "...
{ "lo": 3.81, "mid": 6.33, "hi": 9.49 }
f615cc
antilemma_k3_v1_784195855_7594
Let $S$ be the set of all positive integers $x$ such that $x^2 - 2087x - 2088 = 0$. Let $x = \sum_{d \mid s} \phi(d)$, where $s$ is the sum of all elements in $S$, and the sum is taken over all positive divisors $d$ of $s$. Compute the remainder when $8 - x$ is divided by $84311$.
82,232
graphs = [ Graph( let={ "_n": Const(84311), "x": SumOverDivisors(n=SumOverSet(set=SolutionsSet(var=Var(name='x'), condition=Eq(left=Sum(Pow(base=Var(name='x'), exp=Const(value=2)), Mul(Const(value=-2087), Var(name='x')), Const(value=-2088)), right=Const(value=0)))), var='d', expr=Eul...
NT
COMB
COMPUTE
sympy
K13
[ "VIETA_SUM/K3", "K3" ]
78a626
antilemma_k3_v1
null
4
0
[ "K13", "K3", "VIETA_SUM" ]
3
0.002
2026-02-08T09:24:36.444823Z
{ "verified": true, "answer": 82232, "timestamp": "2026-02-08T09:24:36.446338Z" }
82fb94
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 142, "completion_tokens": 617 }, "timestamp": "2026-02-14T03:46:28.165Z", "answer": 82231 }, ...
1
[ { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "DS2", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "K3", "status": "ok" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" }, { "lemma": "V7", "status": "no" }, { ...
{ "lo": -5.49, "mid": 0.22, "hi": 6.51 }
672ddc
antilemma_sum_equals_v1_349078426_403
Let $m = 43$ and $n = 83$. Define $x$ to be the number of ordered pairs $(i, j)$ of positive integers such that $i + j = n$ and $1 \leq i, j \leq 81$. Let $k$ be the number of integers $t$ with $9 \leq t \leq 52$ for which there exist positive integers $a$ and $b$ such that $1 \leq a \leq 8$, $1 \leq b \leq 4$, and $t ...
9,003
graphs = [ Graph( let={ "_m": Const(43), "_n": Const(83), "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(81)), right=Integer...
COMB
GEOM
COMPUTE
sympy
LIN_FORM
[ "LIN_FORM", "COUNT_SUM_EQUALS" ]
a464cd
antilemma_sum_equals_v1
quadratic_mod
3
0
[ "COUNT_SUM_EQUALS", "LIN_FORM" ]
2
0.009
2026-02-08T13:01:19.796658Z
{ "verified": true, "answer": 9003, "timestamp": "2026-02-08T13:01:19.805476Z" }
31a04e
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 254, "completion_tokens": 8924 }, "timestamp": "2026-02-24T16:58:48.474Z", "answer": 9003 }, { "i...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "ok" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "V8", "st...
{ "lo": -2.35, "mid": 1.22, "hi": 4.84 }
6e5190
sequence_lucas_compute_v1_677425708_280
Let $n$ be the number of positive integers $k$ such that $1 \leq k \leq 1539$ and $81$ divides $k$. Let $L_n$ denote the $n$-th Lucas number, defined by $L_1 = 1$, $L_2 = 3$, and $L_m = L_{m-1} + L_{m-2}$ for $m \geq 3$. Compute the remainder when $13994 \cdot L_n$ is divided by $93527$.
79,160
graphs = [ Graph( let={ "_n": Const(93527), "n": CountOverSet(set=SolutionsSet(var=Var("k"), condition=And(Geq(Var("k"), Const(1)), Leq(Var("k"), Const(1539)), Divides(divisor=Const(81), dividend=Var("k"))), domain='positive_integers')), "result": Lucas(arg=Ref(name='n'))...
ALG
NT
COMPUTE
sympy
C2
[ "C2" ]
9685eb
sequence_lucas_compute_v1
null
3
0
[ "C2" ]
1
0.002
2026-02-08T03:12:40.715278Z
{ "verified": true, "answer": 79160, "timestamp": "2026-02-08T03:12:40.716857Z" }
ab0326
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 230, "completion_tokens": 1788 }, "timestamp": "2026-02-08T20:26:16.236Z", "answer": 79160 }, { "...
1
[ { "lemma": "C2", "status": "ok" }, { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "V1", "status": "no" }, { "lemma": "V7", "status":...
{ "lo": -3.46, "mid": 1.03, "hi": 5.49 }
0030f8
sequence_lucas_compute_v1_349078426_771
Let $S$ be the set of all prime numbers $n$ such that $2 \leq n \leq 19$. Let $n$ be the sum of $\phi(d)$ over all positive divisors $d$ of the largest element in $S$. Let $L_n$ denote the $n$-th Lucas number, defined by $L_1 = 1$, $L_2 = 3$, and $L_k = L_{k-1} + L_{k-2}$ for $k \geq 3$. Compute the remainder when $856...
3,007
graphs = [ Graph( let={ "_n": Const(75758), "n": SumOverDivisors(n=MaxOverSet(set=SolutionsSet(var=Var(name='n'), condition=And(Geq(left=Var(name='n'), right=Const(value=2)), Leq(left=Var(name='n'), right=Const(value=19)), IsPrime(arg=Var(name='n'))))), var='d', expr=EulerPhi(n=Var(n...
NT
null
COMPUTE
sympy
MAX_PRIME_BELOW
[ "MAX_PRIME_BELOW/K3" ]
6b6e89
sequence_lucas_compute_v1
null
4
0
[ "K3", "MAX_PRIME_BELOW" ]
2
0.003
2026-02-08T13:17:31.982993Z
{ "verified": true, "answer": 3007, "timestamp": "2026-02-08T13:17:31.986318Z" }
8414c3
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 184, "completion_tokens": 1235 }, "timestamp": "2026-02-15T12:22:29.533Z", "answer": 3007 }, {...
1
[ { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "K3", "status": "ok_later" }, { "lemma": "K5", "statu...
{ "lo": -5.49, "mid": 0.22, "hi": 6.51 }
0aa431
modular_modexp_compute_v1_1742523217_1867
Let $n$ be an integer such that $2 \leq n \leq 242$ and $n$ is prime. Define $e$ to be the maximum value of such $n$. Let $a = 13$ and $m = 21025$. Compute the remainder when $a^e$ is divided by $m$.
9,563
graphs = [ Graph( let={ "_n": Const(2), "a": Const(13), "e": MaxOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Ref("_n")), Leq(Var("n"), Const(242)), IsPrime(Var("n"))))), "m": Const(21025), "result": ModExp(base=Ref("a"), exp=R...
NT
null
COMPUTE
sympy
MAX_PRIME_BELOW
[ "MAX_PRIME_BELOW" ]
dc3ad3
modular_modexp_compute_v1
null
4
0
[ "MAX_PRIME_BELOW" ]
1
0.003
2026-02-08T04:18:46.492692Z
{ "verified": true, "answer": 9563, "timestamp": "2026-02-08T04:18:46.495331Z" }
27a75f
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 182, "completion_tokens": 3007 }, "timestamp": "2026-02-10T16:10:20.537Z", "answer": 9563 }, { "...
1
[ { "lemma": "K13", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "L3b", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok" } ]
{ "lo": -3.42, "mid": 1.86, "hi": 7.55 }
2ce9b0
comb_count_surjections_v1_397696148_1139
Let $n = 7$. Let $k$ be the number of ordered pairs $(i,j)$ of integers such that $1 \leq i \leq 5$, $1 \leq j \leq 5$, and $i + j = 6$. Compute $k! \cdot S(n, k)$, where $S(n, k)$ denotes the Stirling number of the second kind. Find the value of this expression.
16,800
graphs = [ Graph( let={ "n": Const(7), "k": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var("i"), Var("j")]), condition=Eq(Sum(Var("i"), Var("j")), Const(6)), domain=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(5)), right=IntegerRange(start=Const(1), end=Const(5...
COMB
null
COUNT
sympy
COUNT_SUM_EQUALS
[ "COUNT_SUM_EQUALS" ]
75ab0f
comb_count_surjections_v1
null
5
0
[ "COUNT_SUM_EQUALS" ]
1
0.012
2026-02-08T12:22:30.516106Z
{ "verified": true, "answer": 16800, "timestamp": "2026-02-08T12:22:30.528331Z" }
d1f8e1
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 205, "completion_tokens": 871 }, "timestamp": "2026-02-24T15:37:21.064Z", "answer": 16800 }, { "i...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "ok" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", "status":...
{ "lo": -3.84, "mid": -1.67, "hi": 1.32 }
c4fab5_l
modular_product_range_v1_548369836_52
Let $m = 8$ and $n = 2130$. Define $a$ to be the sum of $\phi(d)$ over all positive divisors $d$ of $99$. Let $b$ be the number of positive integers $k$ such that $1 \leq k \leq n$ and $m$ divides $F_k$, where $F_k$ denotes the $k$th Fibonacci number. Compute the remainder when the product $\prod_{i=a}^{b} i$ is divide...
0
NT
null
COMPUTE
sympy
COUNT_FIB_DIVISIBLE
[ "COUNT_FIB_DIVISIBLE", "K3" ]
1c15c3
modular_product_range_v1
null
6
0
[ "COUNT_FIB_DIVISIBLE", "K3" ]
2
0.003
2026-02-08T02:44:41.983706Z
{ "verified": false, "answer": 209, "timestamp": "2026-02-08T02:44:41.986737Z" }
6b92d2
c4fab5
legacy_text
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 222, "completion_tokens": 32768 }, "timestamp": "2026-02-23T16:01:24.118Z", "answer": null }, { ...
1
[ { "lemma": "COUNT_DIVISOR_COUNT", "status": "no" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "ok" }, { "lemma": "K3", "status": "ok" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MAX_VAL", "status": "no" }, { "lemma": "V5", "status": "n...
{ "lo": 3.82, "mid": 5.54, "hi": 7.58 }
0d3173
comb_bell_compute_v1_1431428450_409
Let $n$ be the minimum value of $x + y$ over all ordered pairs $(x, y)$ of positive integers such that $xy = 16$. Let $Q = 11449 - B_n$, where $B_n$ denotes the $n$th Bell number, which counts the number of partitions of a set of size $n$. Compute $Q$.
7,309
graphs = [ Graph( let={ "_n": Const(11449), "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(16)))), expr=Sum(Var("x"), Var("y")))), ...
COMB
null
COMPUTE
sympy
SUM_ARITHMETIC
[ "B3" ]
0cd20d
comb_bell_compute_v1
null
3
0
[ "B3", "SUM_ARITHMETIC" ]
2
0.015
2026-02-08T13:27:01.696907Z
{ "verified": true, "answer": 7309, "timestamp": "2026-02-08T13:27:01.711735Z" }
a98f10
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": 1448 }, "timestamp": "2026-02-24T18:10:48.311Z", "answer": 7309 }, { "i...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", "status": "n...
{ "lo": -3.84, "mid": -1.67, "hi": 1.32 }
619d9b
comb_binomial_compute_v1_784195855_6637
Let $n$ be the number of positive integers less than or equal to 25 such that the sum of the digits of $n$ leaves a remainder of 1 when divided by 2. Let $k = 6$. Compute the remainder when $88834 \times \binom{n}{k}$ is divided by 79263.
16,395
graphs = [ Graph( let={ "_n": Const(2), "n": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), Const(25)), Eq(Mod(value=DigitSum(Var("n")), modulus=Ref("_n")), Const(1))))), "k": Const(6), "result": Binom(n=Ref("n...
ALG
COMB
COMPUTE
sympy
L3B
[ "L3B" ]
cc148f
comb_binomial_compute_v1
null
4
0
[ "L3B" ]
1
0.001
2026-02-08T08:46:22.680336Z
{ "verified": true, "answer": 16395, "timestamp": "2026-02-08T08:46:22.681372Z" }
b94241
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": 32768 }, "timestamp": "2026-02-24T09:58:24.263Z", "answer": 16395 }, { ...
1
[ { "lemma": "L3B", "status": "ok" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" } ]
{ "lo": -2.35, "mid": 1.22, "hi": 4.84 }
3f3dc8
comb_count_surjections_v1_898971024_1536
Let $n = 5$ and $k = 4$. Compute $30634$ times $k!$ multiplied by the number of ways to partition a set of $n$ elements into $k$ nonempty unlabeled subsets, and then take the result modulo $93495$.
59,550
graphs = [ Graph( let={ "n": Const(5), "k": Const(4), "result": Mul(Factorial(Ref("k")), Stirling2(n=Ref(name='n'), k=Ref(name='k'))), "Q": Mod(value=Mul(Const(30634), Ref("result")), modulus=Const(93495)), }, goal=Ref("Q"), ) ]
COMB
null
COUNT
sympy
LIN_FORM
[ "LIN_FORM/COUNT_SUM_EQUALS" ]
8a3f7a
comb_count_surjections_v1
null
3
0
[ "COUNT_SUM_EQUALS", "LIN_FORM" ]
2
0.037
2026-02-08T16:11:32.486622Z
{ "verified": true, "answer": 59550, "timestamp": "2026-02-08T16:11:32.524045Z" }
63aeae
CC BY 4.0
[ { "id": 2, "model": "openai/gpt-oss-120b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 169, "completion_tokens": 743 }, "timestamp": "2026-02-24T20:05:43.137Z", "answer": 59550 }, { "...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "ok_later" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", "status": "no"...
{ "lo": -2.4, "mid": 1.57, "hi": 5.75 }
419c99
comb_binomial_compute_v1_1218484723_4685
Let $k$ be the number of ordered pairs $(a, b)$ of positive integers with $1 \le a, b \le 25$ satisfying \[ 17a^4 + 68a^3b + 102a^2b^2 + 68ab^3 + 17b^4 = 69632. \] Compute $\binom{13}{k}$.
1,716
graphs = [ Graph( let={ "_n": Const(2), "n": Const(13), "k": 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(102), ...
COMB
null
COMPUTE
sympy
POLY4_COUNT
[ "POLY4_COUNT" ]
861d91
comb_binomial_compute_v1
null
4
0
[ "POLY4_COUNT" ]
1
0.002
2026-02-25T06:21:24.193341Z
{ "verified": true, "answer": 1716, "timestamp": "2026-02-25T06:21:24.195382Z" }
6196d1
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": 795 }, "timestamp": "2026-03-29T16:59:30.679Z", "answer": 1716 }, { "id...
1
[ { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "POLY4_COUNT", "status": "ok" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8", "status": "no" }, { "lemma": "V8_SUM", "status": ...
{ "lo": -3.33, "mid": 1.03, "hi": 5.26 }
297e60_n
geo_visible_lattice_v1_1218484723_1886
A digital art canvas has dimensions $n \times n$, where $n = \sum_{k=1}^{18} \varphi(k) \cdot \left\lfloor \frac{18}{k} \right\rfloor$. A pixel at position $(x,y)$ lights up only if $\gcd(x,y) = 1$. How many pixels light up?
17,875
GEOM
GEOM
COUNT
sympy
K2
[ "K2" ]
6897ab
geo_visible_lattice_v1
null
3
null
[ "K2" ]
1
0.597
2026-02-25T03:35:31.752451Z
null
852701
297e60
narrative
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 189, "completion_tokens": 29161 }, "timestamp": "2026-03-30T17:31:52.028Z", "answer": 17875 }, { ...
1
[ { "lemma": "K2", "status": "ok" } ]
{ "lo": 1.5, "mid": 4.69, "hi": 7.23 }
91de3e
antilemma_k3_v1_1520064083_4071
Compute the sum of $\phi(d)$ over all positive divisors $d$ of 31795, where $\phi(d)$ denotes Euler's totient function.
31,795
graphs = [ Graph( let={ "x": SumOverDivisors(n=Const(value=31795), 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
0.001
2026-02-08T06:03:19.369695Z
{ "verified": true, "answer": 31795, "timestamp": "2026-02-08T06:03:19.370370Z" }
654871
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": 5633 }, "timestamp": "2026-02-12T19:16:07.126Z", "answer": 31795 }, {...
1
[ { "lemma": "DS2", "status": "no" }, { "lemma": "K3", "status": "ok" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "MOD_ADD", "status": "no" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, { "lemma": "MOD_MUL", "status": "no" } ]
{ "lo": -3.44, "mid": 1.34, "hi": 6.56 }
4cf72b
comb_count_partitions_v1_865884756_3379
Let $S$ be the set of all ordered pairs $(x,y)$ of positive integers such that $xy = N$, where $N$ is the number of positive integers $n_1$ at most $M$ that are divisible by 5 and relatively prime to 6, and $M$ is the number of integers $t$ with $8 \le t \le 6620$ that can be expressed as $t = 3a + 5b$ for some integer...
28,259
graphs = [ Graph( let={ "_n": Const(5), "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")), CountOverSet(set=SolutionsSet(var=Var("n1"), condi...
NT
COMB
COUNT
sympy
LIN_FORM
[ "LIN_FORM/C5/B3" ]
88b3fd
comb_count_partitions_v1
null
7
0
[ "B3", "C5", "LIN_FORM" ]
3
0.005
2026-02-08T17:19:20.420060Z
{ "verified": true, "answer": 28259, "timestamp": "2026-02-08T17:19:20.425421Z" }
6087d8
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 234, "completion_tokens": 6566 }, "timestamp": "2026-02-17T23:42:34.863Z", "answer": 28259 }, ...
1
[ { "lemma": "B3", "status": "ok_later" }, { "lemma": "C5", "status": "ok_later" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "K13", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "LTE_DIFF", "status": "no" ...
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
5beb1c
sequence_count_fib_divisible_v1_1918700295_3678
Let $d$ be the minimum value of $x + y$ over all ordered pairs $(x, y)$ of positive integers such that $x \cdot y = 9$. Define $A$ to be the set of all positive integers $n$ such that $1 \leq n \leq 266$ and the $n$-th Fibonacci number is divisible by $d$. Let $r = |A|$, the number of elements in $A$. Compute the remai...
30,800
graphs = [ Graph( let={ "_n": Const(52838), "upper": Const(266), "d": 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(9)))), ...
NT
null
COUNT
sympy
B3
[ "B3" ]
0cd20d
sequence_count_fib_divisible_v1
null
4
0
[ "B3" ]
1
0.064
2026-02-08T08:49:38.733603Z
{ "verified": true, "answer": 30800, "timestamp": "2026-02-08T08:49:38.797243Z" }
4e0183
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 159, "completion_tokens": 1687 }, "timestamp": "2026-02-13T21:44:59.048Z", "answer": 30800 }, ...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "K13", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" }, { "lemma": "MOD_MUL", "status": "no" }, { "lemma": "V8", "status": "no" } ]
{ "lo": -3.51, "mid": 1.33, "hi": 6.56 }
5ee09f
nt_count_gcd_equals_v1_1520064083_1368
Let $S$ be the set of all ordered pairs $(x_1, x_2)$ of positive odd integers such that $x_1 + x_2 = 19208$. Let $N$ be the number of elements in $S$. Compute the number of positive integers $n$ such that $1 \leq n \leq N$ and $\gcd(n, 153) = 17$.
376
graphs = [ Graph( let={ "upper": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var("x1"), Var("x2")]), condition=And(IsPositive(arg=Var(name='x1')), IsPositive(arg=Var(name='x2')), IsOdd(arg=Var(name='x1')), IsOdd(arg=Var(name='x2')), Eq(Sum(Var("x1"), Var("x2")), Const(19208))))), ...
NT
null
COUNT
sympy
COMB1
[ "COMB1" ]
567f58
nt_count_gcd_equals_v1
null
6
0
[ "COMB1" ]
1
4.522
2026-02-08T03:56:56.218911Z
{ "verified": true, "answer": 376, "timestamp": "2026-02-08T03:57:00.740987Z" }
d78726
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 197, "completion_tokens": 1166 }, "timestamp": "2026-02-10T16:13:26.208Z", "answer": 376 }, { "id...
1
[ { "lemma": "COMB1", "status": "ok" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "MAX_VAL", "status": "no" }, { "lemma": "MOD_ADD", "status": "no" }, { "lemma": "V8", "status": "no" }, { "lemma": "VAL_SUM_EQ", "status": "no" } ]
{ "lo": -3.46, "mid": 1.03, "hi": 5.49 }
3fe39d
alg_poly_orbit_count_v1_1218484723_5358
For each non-negative integer $a$ with $0 \leq a \leq 43035$, define \begin{align*} N &= 3a^3 - 3 \bmod 53, \\ M &= 3N^3 - 3 \bmod 53, \\ R &= 3M^3 - 3 \bmod 53, \\ S &= 3R^3 - 3 \bmod 53, \\ T &= 3S^3 - 3 \bmod 53. \end{align*} Let $Q$ be the number of such $a$ for which $T = a$, but $N \neq a$, $M \neq a$, $R \neq a$...
4,060
graphs = [ Graph( let={ "p1": Mod(value=Sum(Mul(Const(3), Pow(Var("a"), Const(3))), Const(-3)), modulus=Const(53)), "p2": Mod(value=Sum(Mul(Const(3), Pow(Ref("p1"), Const(3))), Const(-3)), modulus=Const(53)), "p3": Mod(value=Sum(Mul(Const(3), Pow(Ref("p2"), Const(3))), Co...
ALG
null
COUNT
sympy
POLY_ORBIT_COUNT
[ "POLY_ORBIT_COUNT" ]
4ad965
alg_poly_orbit_count_v1
null
6
null
[ "POLY_ORBIT_COUNT" ]
1
0.023
2026-02-25T06:57:10.633870Z
{ "verified": true, "answer": 4060, "timestamp": "2026-02-25T06:57:10.656751Z" }
d94fdf
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 283, "completion_tokens": 8220 }, "timestamp": "2026-03-29T20:43:23.325Z", "answer": 5 }, { "i...
1
[ { "lemma": "POLY_ORBIT_COUNT", "status": "ok" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" } ]
{ "lo": 3.81, "mid": 6.33, "hi": 9.49 }
62cae2
algebra_poly_eval_v1_1520064083_5592
Let $n = 11$. Define $$ Q = 6n^4 + 2n^k + 3n^2 - 2n - 2, $$ where $k$ is the smallest divisor of $75$ that is at least $2$. Compute the value of $Q$.
90,847
graphs = [ Graph( let={ "_n": Const(6), "n": Const(11), "result": Sum(Mul(Ref("_n"), Pow(Ref("n"), Const(4))), Mul(Const(2), Pow(Ref("n"), MinOverSet(set=SolutionsSet(var=Var("d"), condition=And(Geq(Var("d"), Const(2)), Divides(divisor=Var("d"), dividend=Const(75))))))), ...
NT
null
COMPUTE
sympy
MIN_PRIME_FACTOR
[ "MIN_PRIME_FACTOR" ]
bc3776
algebra_poly_eval_v1
null
3
0
[ "MIN_PRIME_FACTOR" ]
1
0.003
2026-02-08T07:26:56.023178Z
{ "verified": true, "answer": 90847, "timestamp": "2026-02-08T07:26:56.026292Z" }
f1763b
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 110, "completion_tokens": 571 }, "timestamp": "2026-02-13T10:24:10.420Z", "answer": 90847 }, {...
1
[ { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "K15", "status": "no" }, { "lemma": "L3c", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "MIN_PRIME_FACTOR", "status": "ok" }, { "lemma": "MOD_FACTORIAL", "status": ...
{ "lo": -5.55, "mid": -3.01, "hi": 0.32 }
f8dc48
antilemma_sum_equals_v1_153355830_2237
Let $n = 31$. Let $x$ be the number of ordered pairs $(i, j)$ of integers such that $1 \leq i \leq 30$, $1 \leq j \leq 30$, and $i + j = n$. Let $Q$ be the Bell number $B_r$, where $r$ is the remainder when $|x|$ is divided by $11$. Compute $Q$.
4,140
graphs = [ Graph( let={ "_n": Const(31), "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(30)), right=IntegerRange(start=Const(1), end=Con...
COMB
GEOM
COMPUTE
sympy
COUNT_SUM_EQUALS
[ "COUNT_SUM_EQUALS" ]
75ab0f
antilemma_sum_equals_v1
null
3
0
[ "COUNT_SUM_EQUALS" ]
1
0.011
2026-02-08T07:00:12.023136Z
{ "verified": true, "answer": 4140, "timestamp": "2026-02-08T07:00:12.033889Z" }
51e61e
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 203, "completion_tokens": 1278 }, "timestamp": "2026-02-24T07:31:14.565Z", "answer": 4140 }, { "i...
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_SU...
{ "lo": -5.97, "mid": -3.97, "hi": -1.93 }
7a62e4
nt_sum_totient_over_divisors_v1_151522320_48
Let $N$ be the number of positive integers $p$ for which there exists a positive integer $q$ such that \[pq = 18,\quad \gcd(p,q)=1,\quad p<q.\] Let $n=27619$, and define \[R = \sum_{d\mid n} \varphi(d),\] where $\varphi$ is Euler's totient function. Let $T$ be the set of all integers $t$ for which there exist integer...
21,147
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=18)), Eq(left=GCD(a=Var(name='p'), b=Var(name='q')), right=Const(value=1)), ...
NT
COMB
COMPUTE
sympy
COPRIME_PAIRS
[ "COPRIME_PAIRS/LIN_FORM/MAX_PRIME_BELOW" ]
8f4fb8
nt_sum_totient_over_divisors_v1
bell_mod
7
0
[ "COPRIME_PAIRS", "LIN_FORM", "MAX_PRIME_BELOW" ]
3
0.004
2026-02-08T02:56:17.370930Z
{ "verified": true, "answer": 21147, "timestamp": "2026-02-08T02:56:17.375402Z" }
12edf0
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 413, "completion_tokens": 1686 }, "timestamp": "2026-02-10T11:57:21.393Z", "answer": 21147 }, { "...
1
[ { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok_later" }, { "lemma": "LTE_SUM", "status": "no" ...
{ "lo": -1.94, "mid": 0.57, "hi": 2.67 }
30e542
algebra_poly_eval_v1_1218484723_1679
Let $R$ be the number of ordered pairs $(a, b)$ of positive integers with $1 \le a, b \le 15$ such that $5b^2 - 2ab + 34a^2 = 1690$. Let $S$ be the set of integers $t1$ for which there exist integers $a, b$ with $1 \le a \le 163$, $1 \le b \le 19$, $18 \le t1 \le 918$, and $t1 = 4a + 14b$. Let $U = |S|$. Let $t$ be the...
30,079
graphs = [ Graph( let={ "_d": Const(15), "_m": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var("a"), Var("b")]), condition=And(Geq(Var("a"), Const(1)), Leq(Var("a"), Ref("_d")), Geq(Var("b"), Const(1)), Leq(Var("b"), Const(15)), Eq(Sum(Mul(Const(5), Pow(Var("b"), Const(2))), Mu...
ALG
null
COMPUTE
sympy
LIN_FORM
[ "LIN_FORM/QF_PSD_DISTINCT", "QF_PSD_COUNT/LIN_FORM" ]
23714a
algebra_poly_eval_v1
null
6
0
[ "LIN_FORM", "QF_PSD_COUNT", "QF_PSD_DISTINCT" ]
3
0.015
2026-02-25T03:22:03.546457Z
{ "verified": true, "answer": 30079, "timestamp": "2026-02-25T03:22:03.561820Z" }
81dbe5
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 339, "completion_tokens": 5601 }, "timestamp": "2026-03-29T00:46:17.070Z", "answer": 30079 }, { "...
1
[ { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" }, { "lemma": "QF_PSD_COUNT", "status": "ok" }, { "lemma": "QF_PSD_DISTINCT", "status": "ok_later" } ]
{ "lo": -2.47, "mid": 1.2, "hi": 4.81 }
24cb4f
sequence_count_fib_divisible_v1_865884756_602
Let $d$ be the largest prime number $n$ such that $2 \leq n \leq 18$. Determine the value of the number of positive integers $n_1$ less than or equal to 983 for which $d$ divides the $n_1$-th Fibonacci number.
109
graphs = [ Graph( let={ "_n": Const(2), "upper": Const(983), "d": MaxOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Ref("_n")), Leq(Var("n"), Const(18)), IsPrime(Var("n"))))), "result": CountOverSet(set=SolutionsSet(var=Var("n1"), condition...
NT
null
COUNT
sympy
ONE_PHI_2
[ "MAX_PRIME_BELOW" ]
dc3ad3
sequence_count_fib_divisible_v1
null
6
0
[ "MAX_PRIME_BELOW", "ONE_PHI_2" ]
2
0.188
2026-02-08T15:31:38.733605Z
{ "verified": true, "answer": 109, "timestamp": "2026-02-08T15:31:38.922015Z" }
819738
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 108, "completion_tokens": 2234 }, "timestamp": "2026-02-16T07:42:01.393Z", "answer": 109 }, { ...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "L3b", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok" }, { "lemma": "MOD_MUL", "status": "no" }, { "lemma": "VAL_SUM_EQ", "status": "no" } ]
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
5d2a6a
algebra_poly_eval_v1_124444284_6703
Let $k = 8$. Define $s$ to be the maximum value of $xy$ over all ordered pairs $(x, y)$ of positive integers such that $x + y = 4$. Let $r$ be the value of the expression $$ \frac{81 \cdot k^s + 1341 \cdot k^3 - 660 \cdot k^2 - 825 \cdot k - 153}{25 \cdot 75}. $$ Compute the remainder when $10085r$ is divided by $78026...
64,229
graphs = [ Graph( let={ "_n": Const(2), "k": Const(8), "result": Div(Sum(Mul(Const(81), Pow(Ref("k"), 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(Va...
NT
null
COMPUTE
sympy
COUNT_CARTESIAN
[ "COUNT_CARTESIAN", "B1" ]
36c577
algebra_poly_eval_v1
null
6
0
[ "B1", "COUNT_CARTESIAN" ]
2
0.003
2026-02-08T08:35:45.290680Z
{ "verified": true, "answer": 64229, "timestamp": "2026-02-08T08:35:45.294004Z" }
a07c28
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 159, "completion_tokens": 1249 }, "timestamp": "2026-02-13T19:46:17.346Z", "answer": 64229 }, ...
1
[ { "lemma": "B1", "status": "ok" }, { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_CARTESIAN", "status": "ok" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" }, { "lemma": "MOD_ADD", "status": "no" }, { "lemma": "V3", "status": "no" }, { ...
{ "lo": -3.44, "mid": 1.34, "hi": 6.56 }
0585ec
nt_min_coprime_above_v1_397696148_742
Let $S$ be the set of integers $t$ such that $7 \leq t \leq 205$ and there exist integers $a$ and $b$ with $1 \leq a \leq 25$, $1 \leq b \leq 40$, and $t = 5a + 2b$. Let $m$ be the number of elements in $S$. Determine the value of the smallest integer $n$ such that $71631 < n \leq 71836$ and $\gcd(n, m) = 1$.
71,632
graphs = [ Graph( let={ "start": Const(71631), "upper": Const(71836), "modulus": 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(n...
NT
null
EXTREMUM
sympy
LIN_FORM
[ "LIN_FORM" ]
7b2633
nt_min_coprime_above_v1
null
4
0
[ "LIN_FORM" ]
1
0.043
2026-02-08T11:43:03.736127Z
{ "verified": true, "answer": 71632, "timestamp": "2026-02-08T11:43:03.779197Z" }
a2b9c9
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": 3557 }, "timestamp": "2026-02-14T18:03:28.579Z", "answer": 71632 }, ...
1
[ { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "V3", "status": "no" }, { "lemma": "V5", "status": "no" }, { "lemma": "V8", "status": "no" }, { "lemma": "V8_SUM", "status": "no" } ]
{ "lo": -5.14, "mid": 0.32, "hi": 6.51 }
0a0b22
alg_qf_psd_min_v1_1419126231_530
Find the minimum value of $19850a^2 + 39700ab + 23026b^2$ over all ordered pairs $(a, b)$ of positive integers with $1 \leq a \leq 99$ and $1 \leq b \leq B$, where $B$ is the number of integers $v$ such that $4 \leq v \leq C$ and there exist integers $a, b$ with $1 \leq a, b \leq 14$ satisfying $8a^2 + 4b^2 - 8ab = v$,...
82,576
graphs = [ Graph( let={ "_m": Const(2), "_n": Const(2), "result": MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var("a"), Var("b")]), condition=And(Geq(Var("a"), Const(1)), Leq(Var("a"), Const(99)), Geq(Var("b"), Const(1)), Leq(Var("b"), CountOverSet(set=...
ALG
null
COMPUTE
sympy
QF_PSD_COUNT_LEQ
[ "QF_PSD_COUNT_LEQ/QF_PSD_DISTINCT" ]
0cf842
alg_qf_psd_min_v1
null
6
0
[ "QF_PSD_COUNT_LEQ", "QF_PSD_DISTINCT" ]
2
0.339
2026-02-25T10:03:40.358839Z
{ "verified": true, "answer": 82576, "timestamp": "2026-02-25T10:03:40.697640Z" }
48d1e3
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 309, "completion_tokens": 17850 }, "timestamp": "2026-03-30T08:54:05.197Z", "answer": 82576 }, { ...
1
[ { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" }, { "lemma": "QF_PSD_COUNT_LEQ", "status": "ok" }, { "lemma": "QF_PSD_DISTINCT", "status": "ok_later" } ]
{ "lo": 1.5, "mid": 4.69, "hi": 7.23 }
b1cb55
comb_count_permutations_fixed_v1_1918700295_2740
Let $\_n = 5$. Define $n$ to be the sum of all positive integers $m$ such that $1 \leq m \leq \_n$ and $$m \equiv \sum_{k=0}^{7} (-1)^k \binom{7}{k} \pmod{5}.$$ Let $k = 0$ and define $\text{result} = \binom{n}{k} \cdot !(n - k)$, where $!r$ denotes the number of derangements of $r$ elements. Compute the remainder when...
20,540
graphs = [ Graph( let={ "_n": Const(5), "n": SumOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), Ref("_n")), Eq(Mod(value=Var("n"), modulus=Const(5)), Summation(var="k", start=Const(0), end=Const(7), expr=Mul(Pow(Const(-1), Var("k")), Binom(...
COMB
null
COUNT
sympy
BINOMIAL_ALTERNATING
[ "BINOMIAL_ALTERNATING", "SUM_DIVISIBLE" ]
7f5b8a
comb_count_permutations_fixed_v1
null
5
0
[ "BINOMIAL_ALTERNATING", "SUM_DIVISIBLE" ]
2
0.152
2026-02-08T08:11:08.693902Z
{ "verified": true, "answer": 20540, "timestamp": "2026-02-08T08:11:08.845940Z" }
f6c288
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 260, "completion_tokens": 1196 }, "timestamp": "2026-02-24T09:01:23.539Z", "answer": 20540 }, { "...
1
[ { "lemma": "BINOMIAL_ALTERNATING", "status": "ok" }, { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "no" }, { "lemma": "SUM_DIVISIBLE", "status": "ok" }, { "lemma": "V7", ...
{ "lo": -2.35, "mid": 1.22, "hi": 4.84 }
44af55
modular_sum_quadratic_residues_v1_1520064083_6458
Let $p = 181$. Let $N$ be the number of positive integers $p'$ such that there exists a positive integer $q$ with $p'q = 750$, $\gcd(p', q) = 1$, and $p' < q$. Compute $\frac{p(p-1)}{N}$. Determine the remainder when $44121$ times this value is divided by $82394$.
45,311
graphs = [ Graph( let={ "p": Const(181), "result": Div(Mul(Ref("p"), Sub(Ref("p"), Const(1))), CountOverSet(set=SolutionsSet(var=Var("p"), condition=And(IsPositive(arg=Var(name='p')), Exists(var=Var(name='q'), condition=And(Eq(left=Mul(Var(name='p'), Var(name='q')), right=Const(value...
NT
null
SUM
sympy
COPRIME_PAIRS
[ "COPRIME_PAIRS" ]
2bb3aa
modular_sum_quadratic_residues_v1
null
4
0
[ "COPRIME_PAIRS" ]
1
0.001
2026-02-08T08:05:52.071935Z
{ "verified": true, "answer": 45311, "timestamp": "2026-02-08T08:05:52.073315Z" }
b298d6
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 132, "completion_tokens": 3045 }, "timestamp": "2026-02-13T15:06:34.037Z", "answer": 45311 }, ...
1
[ { "lemma": "COPRIME_PAIRS", "status": "ok" }, { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "L3c", "status": "no" }, { "lemma": "MOD_ADD", "status": "no" }, { "lemma": "VAL_SUM_EQ", "status": "no" } ]
{ "lo": -3.44, "mid": 1.34, "hi": 6.56 }
6d9249
algebra_poly_eval_v1_784195855_7947
Let $n = 7$ and let $\_n = 2$. Define $$ \text{result} = 7 \cdot n^{\_n} + \left( \sum_{k=1}^{4} \phi(k) \left\lfloor \frac{4}{k} \right\rfloor \right) \cdot n + 6, $$ where $\phi(k)$ denotes Euler's totient function. Compute the value of $\text{result}$.
419
graphs = [ Graph( let={ "_n": Const(2), "n": Const(7), "result": Sum(Mul(Const(7), Pow(Ref("n"), Ref("_n"))), Mul(Summation(var="k", start=Const(1), end=Const(4), expr=Mul(EulerPhi(n=Var("k")), Floor(Div(Const(4), Var("k"))))), Ref("n")), Const(6)), }, goa...
NT
null
COMPUTE
sympy
K2
[ "K2" ]
6897ab
algebra_poly_eval_v1
null
3
0
[ "K2" ]
1
0.001
2026-02-08T09:38:00.270660Z
{ "verified": true, "answer": 419, "timestamp": "2026-02-08T09:38:00.271779Z" }
9db2f1
CC BY 4.0
[ { "id": 8, "model": "mathstral", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 163, "completion_tokens": 444 }, "timestamp": "2026-02-15T20:47:34.472Z", "answer": 419 }, { "id": 11, ...
2
[ { "lemma": "K13", "status": "no" }, { "lemma": "K18", "status": "no" }, { "lemma": "K2", "status": "ok" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "MOD_SUB", "status": "no" }, { "lemma": "V8_SUM", "status": "no" } ]
{ "lo": -10, "mid": -7.71, "hi": -5.43 }
3ceef0
comb_sum_binomial_row_v1_2051736721_1520
Let $n$ be the number of positive integers $k$ such that $1 \leq k \leq 1260$ and $90$ divides $k$. Define $r = 2^n$. Compute the value of $$r + \phi(|r| + 1) + \tau(|r| + 1),$$ where $\phi$ denotes Euler's totient function and $\tau(m)$ denotes the number of positive divisors of $m$.
28,936
graphs = [ Graph( let={ "_n": Const(1260), "n": CountOverSet(set=SolutionsSet(var=Var("k"), condition=And(Geq(Var("k"), Const(1)), Leq(Var("k"), Ref("_n")), Divides(divisor=Const(90), dividend=Var("k"))), domain='positive_integers')), "result": Pow(Const(2), Ref("n")), ...
NT
null
SUM
sympy
C2
[ "C2" ]
9685eb
comb_sum_binomial_row_v1
null
3
0
[ "C2" ]
1
0.002
2026-02-08T16:05:07.340973Z
{ "verified": true, "answer": 28936, "timestamp": "2026-02-08T16:05:07.343378Z" }
35686d
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 140, "completion_tokens": 990 }, "timestamp": "2026-02-16T20:44:24.772Z", "answer": 28936 }, {...
1
[ { "lemma": "C2", "status": "ok" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "K14", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" }, { "lemma": "V3", "status": "no" } ]
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
de72df
comb_factorial_compute_v1_124444284_7018
Let $n$ be the smallest integer greater than or equal to 2 that divides 77. Compute $n!$.
5,040
graphs = [ Graph( let={ "_n": Const(2), "n": MinOverSet(set=SolutionsSet(var=Var("d"), condition=And(Geq(Var("d"), Ref("_n")), Divides(divisor=Var("d"), dividend=Const(77))))), "result": Factorial(Ref("n")), "Q": Ref("result"), }, goal=Ref("Q")...
NT
null
COMPUTE
sympy
MIN_PRIME_FACTOR
[ "MIN_PRIME_FACTOR" ]
bc3776
comb_factorial_compute_v1
null
2
0
[ "MIN_PRIME_FACTOR" ]
1
0.001
2026-02-08T08:46:04.116728Z
{ "verified": true, "answer": 5040, "timestamp": "2026-02-08T08:46:04.117761Z" }
f37a5b
CC BY 4.0
[ { "id": 8, "model": "mathstral", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 82, "completion_tokens": 437 }, "timestamp": "2026-02-15T20:21:35.118Z", "answer": 5040 }, { "id": 11, ...
2
[ { "lemma": "C2", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" }, { "lemma": "MIN_PRIME_FACTOR", "status": "ok" }, { "lemma": "MOD_SUB", "status": "no" }, { "lemma": "V1", "status": "no" } ]
{ "lo": -10, "mid": -7.73, "hi": -5.46 }
dcd4a7
nt_min_coprime_above_v1_397696148_869
Let $S$ be the set of all ordered pairs $(x,y)$ of positive integers such that $xy = 1000000$. Let $T$ be the set of all values $x + y$ where $(x,y) \in S$. Let $a$ be the minimum value in $T$. Let $U$ be the set of all integers $n$ such that $a < n \leq 2303$ and $\gcd(n, 293) = 1$. Let $b$ be the minimum element of ...
16,768
graphs = [ Graph( let={ "start": 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")))), "upper": Co...
NT
null
EXTREMUM
sympy
B3
[ "B3" ]
0cd20d
nt_min_coprime_above_v1
null
5
0
[ "B3" ]
1
0.054
2026-02-08T11:47:19.805328Z
{ "verified": true, "answer": 16768, "timestamp": "2026-02-08T11:47:19.859760Z" }
9e76a2
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": 1079 }, "timestamp": "2026-02-14T21:00:15.314Z", "answer": 16768 }, ...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "L3c", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "MOD_FACTORIAL", "status": "no" }, { "lemma": "VAL_SUM_EQ", "status": ...
{ "lo": -6.96, "mid": -4.56, "hi": -1.46 }
106e2c
sequence_lucas_compute_v1_1915831931_2629
Let $m = 2$. Define $D$ as the set of all divisors $d$ of $5538101$ such that $d \geq m$. Let $n$ be the smallest element of $D$. Define $s = \sum_{d_1 \mid n} \phi(d_1)$, where $\phi$ denotes Euler's totient function. Compute the $s$-th Lucas number.
9,349
graphs = [ Graph( let={ "_m": Const(2), "_n": MinOverSet(set=SolutionsSet(var=Var("d"), condition=And(Geq(Var("d"), Ref("_m")), Divides(divisor=Var("d"), dividend=Const(5538101))))), "n": SumOverDivisors(n=Ref(name='_n'), var='d1', expr=EulerPhi(n=Var(name='d1'))), ...
NT
null
COMPUTE
sympy
MIN_PRIME_FACTOR
[ "MIN_PRIME_FACTOR/K3" ]
54b4a9
sequence_lucas_compute_v1
null
6
0
[ "K3", "MIN_PRIME_FACTOR" ]
2
0.002
2026-02-08T17:00:39.347606Z
{ "verified": true, "answer": 9349, "timestamp": "2026-02-08T17:00:39.349506Z" }
b63b06
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 132, "completion_tokens": 1670 }, "timestamp": "2026-02-17T17:13:17.855Z", "answer": 9349 }, {...
1
[ { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "DS2", "status": "no" }, { "lemma": "K16", "status": "no" }, { "lemma": "K18", "status": "no" }, { "lemma": "K3", "status": "ok_later" }, { "lemma": "MAX_PRIME_BELOW", "status": "no" ...
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
a7602a
geo_count_lattice_rect_v1_784195855_4260
Let $a = 136$ and $b = 433$. Define $R$ to be the rectangle $[0, a] \times [0, b]$. Compute the number of lattice points $(x, y)$ with integer coordinates that lie inside or on the boundary of $R$. Let this count be $L$. Find the remainder when $1600 - L$ is divided by $89369$.
31,511
graphs = [ Graph( let={ "a": Const(136), "b": Const(433), "result": LatticePointsRect(a=Ref(name='a'), b=Ref(name='b')), "_c": Const(1600), "Q": Mod(value=Sub(Ref("_c"), Ref("result")), modulus=Const(89369)), }, goal=Ref("Q"), )...
GEOM
null
COUNT
sympy
[]
geo_count_lattice_rect_v1
null
4
0
null
null
0.002
2026-02-08T06:56:49.507954Z
{ "verified": true, "answer": 31511, "timestamp": "2026-02-08T06:56:49.509747Z" }
354e1a
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 195, "completion_tokens": 614 }, "timestamp": "2026-02-24T07:27:56.209Z", "answer": 31511 }, { "i...
1
[]
{ "lo": -5.97, "mid": -3.97, "hi": -1.93 }
2e73d1
geo_count_lattice_rect_v1_1248542787_127
Let $a = 300$ and $b = 88$. Compute the number of lattice points $(x, y)$ such that $0 \leq x \leq a$ and $0 \leq y \leq b$. Find the value of this count.
26,789
graphs = [ Graph( let={ "a": Const(300), "b": Const(88), "result": LatticePointsRect(a=Ref(name='a'), b=Ref(name='b')), }, goal=Ref("result"), ) ]
GEOM
null
COUNT
sympy
[]
geo_count_lattice_rect_v1
null
2
0
null
null
0.001
2026-02-08T02:57:56.324818Z
{ "verified": true, "answer": 26789, "timestamp": "2026-02-08T02:57:56.326188Z" }
8a9f9c
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 171, "completion_tokens": 246 }, "timestamp": "2026-02-09T00:26:54.116Z", "answer": 26789 }, { "i...
1
[]
{ "lo": -9.23, "mid": -6.17, "hi": -4.06 }
f2ecda
sequence_count_fib_divisible_v1_2051736721_488
Let $d = 13$. Determine the number of positive integers $n$ such that $1 \leq n \leq 882$ and $d$ divides the $n$-th Fibonacci number.
126
graphs = [ Graph( let={ "upper": Const(882), "d": Const(13), "result": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Const(1)), Leq(Var("n"), Ref("upper")), Divides(divisor=Ref("d"), dividend=Fibonacci(arg=Var(name='n')))))), }, g...
NT
null
COUNT
sympy
LIN_FORM
[ "LIN_FORM" ]
7b2633
sequence_count_fib_divisible_v1
null
5
0
[ "LIN_FORM" ]
1
0.089
2026-02-08T15:28:14.458476Z
{ "verified": true, "answer": 126, "timestamp": "2026-02-08T15:28:14.547291Z" }
aec5b8
CC BY 4.0
[ { "id": 5, "model": "deepseek-ai/DeepSeek-V3.2", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 88, "completion_tokens": 2156 }, "timestamp": "2026-02-16T06:41:30.136Z", "answer": 126 }, { ...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "K13", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "V8", "status": "no" }, { "lemma": "VAL_SUM_EQ", "status": "no" } ]
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }
e9a330
comb_count_surjections_v1_1431428450_1428
Let $n$ be the number of ordered pairs $(i, j)$ of positive integers such that $1 \leq i \leq 7$, $1 \leq j \leq 7$, and $i + j = 7$. Let $k = 5$. Compute $k! \cdot S(n, k)$, where $S(n, k)$ denotes the Stirling number of the second kind.
1,800
graphs = [ Graph( let={ "n": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var("i"), Var("j")]), condition=Eq(Sum(Var("i"), Var("j")), Const(7)), domain=CartesianProduct(left=IntegerRange(start=Const(1), end=Const(7)), right=IntegerRange(start=Const(1), end=Const(7))))), "k": Con...
COMB
null
COUNT
sympy
COUNT_SUM_EQUALS
[ "COUNT_SUM_EQUALS" ]
75ab0f
comb_count_surjections_v1
null
3
0
[ "COUNT_SUM_EQUALS" ]
1
0.052
2026-02-08T14:05:52.718968Z
{ "verified": true, "answer": 1800, "timestamp": "2026-02-08T14:05:52.770785Z" }
807f8c
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 200, "completion_tokens": 772 }, "timestamp": "2026-02-24T19:45:51.441Z", "answer": 1800 }, { "id...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "COUNT_SUM_EQUALS", "status": "ok" }, { "lemma": "V7", "status": "no" }, { "lemma": "V8_SUM", "status"...
{ "lo": -4.1, "mid": -1.76, "hi": 1.26 }
8aaaeb
comb_count_surjections_v1_1419126231_1724
Let $k = 4$ and $N = k! \cdot S(7, k)$, where $S(7, k)$ denotes the Stirling number of the second kind. Find the remainder when $196 - N$ is divided by $50385$.
42,181
graphs = [ Graph( let={ "n": Const(7), "k": Const(4), "result": Mul(Factorial(Ref("k")), Stirling2(n=Ref(name='n'), k=Ref(name='k'))), "Q": Mod(value=Sub(Const(196), Ref("result")), modulus=Const(50385)), }, goal=Ref("Q"), ) ]
COMB
null
COUNT
sympy
K3
[ "K3/BINOMIAL_ALTERNATING" ]
49270b
comb_count_surjections_v1
negation_mod
3
0
[ "BINOMIAL_ALTERNATING", "K3" ]
2
0.029
2026-02-25T11:14:46.517101Z
{ "verified": true, "answer": 42181, "timestamp": "2026-02-25T11:14:46.545885Z" }
de8631
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 167, "completion_tokens": 818 }, "timestamp": "2026-03-30T13:38:02.614Z", "answer": 42181 }, { "i...
1
[ { "lemma": "BINOMIAL_ALTERNATING", "status": "ok_later" }, { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "K3", "status": "ok" }, { "lemm...
{ "lo": -5.37, "mid": 0.23, "hi": 5.22 }
1deba4
alg_poly3_sum_v1_1218484723_7342
Find the remainder when $$\sum_{\substack{a=1 \\ b=1 \\ c=1}}^{31} \left( -480abc + \min_{\substack{a_1=1 \\ b_1=1}}^{23} (58a_1b_1 + 26a_1^2 + 41b_1^2) \cdot a^3 + 240ac^2 + 390bc^2 + 240ab^2 + 300a^2b + 125c^3 - 300a^2c - 120b^2c + 73b^3 \right)$$ is divided by $84873$.
44,087
graphs = [ Graph( let={ "_n": Const(23), "result": Mod(value=SumOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(elements=[Var("a"), Var("b"), Var("c")]), condition=And(Geq(Var("a"), Const(1)), Leq(Var("a"), Const(31)), Geq(Var("b"), Const(1)), Leq(Var("b"), Const(31)), Geq(Var("c")...
ALG
null
COMPUTE
sympy
QF_PSD_MIN
[ "QF_PSD_MIN" ]
c66ce3
alg_poly3_sum_v1
null
5
0
[ "QF_PSD_MIN" ]
1
0.111
2026-02-25T08:44:58.270441Z
{ "verified": true, "answer": 44087, "timestamp": "2026-02-25T08:44:58.381509Z" }
a4a533
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 257, "completion_tokens": 6364 }, "timestamp": "2026-03-30T04:06:38.550Z", "answer": 13232 }, { ...
1
[ { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" }, { "lemma": "QF_PSD_MIN", "status": "ok" } ]
{ "lo": 1.53, "mid": 4.7, "hi": 7.23 }
135b5d
alg_poly_orbit_count_v1_1218484723_1361
Define a sequence recursively by $N = (a^2 + a + 5) \bmod 17$, $M = (N^2 + N + 5) \bmod 17$, $R = (M^2 + M + 5) \bmod 17$, $S = (R^2 + R + 5) \bmod 17$, $T = (S^2 + S + 5) \bmod 17$, $K = (T^2 + T + 5) \bmod 17$. Find the number of integers $a$ with $0 \le a \le 662$ such that $K = a$, but $a$ does not appear in the se...
234
graphs = [ Graph( let={ "p1": Mod(value=Sum(Pow(Var("a"), Const(2)), Var("a"), Const(5)), modulus=Const(17)), "p2": Mod(value=Sum(Pow(Ref("p1"), Const(2)), Ref("p1"), Const(5)), modulus=Const(17)), "p3": Mod(value=Sum(Pow(Ref("p2"), Const(2)), Ref("p2"), Const(5)), modulu...
ALG
null
COUNT
sympy
POLY_ORBIT_COUNT
[ "POLY_ORBIT_COUNT" ]
4ad965
alg_poly_orbit_count_v1
null
6
null
[ "POLY_ORBIT_COUNT" ]
1
0.037
2026-02-25T03:04:46.734838Z
{ "verified": true, "answer": 234, "timestamp": "2026-02-25T03:04:46.772111Z" }
8a942b
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 276, "completion_tokens": 3950 }, "timestamp": "2026-03-10T06:38:15.272Z", "answer": 234 }, { "id...
1
[ { "lemma": "POLY_ORBIT_COUNT", "status": "ok" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" } ]
{ "lo": 2.68, "mid": 4.77, "hi": 6.68 }
9e9fdd
algebra_quadratic_discriminant_v1_601307018_4336
Let $m = \min\{ |x - y| : x > 0,\, y > 0,\, x \cdot y = 1891 \}$. Let $c$ be the number of ordered pairs $(a_1, b_1)$ of positive integers with $1 \leq a_1 \leq m$ and $1 \leq b_1 \leq 30$ satisfying $$ 512a_1^4 - 2048a_1^3b_1 + 3072a_1^2b_1^2 - 2048a_1b_1^3 + 512b_1^4 = 320000. $$ Compute $5^2 - 4(-1) \cdot c$.
225
graphs = [ Graph( let={ "_n": Const(4), "a": Const(-1), "b": Const(5), "c": CountOverSet(set=SolutionsSet(var=Tuple(elements=[Var("a1"), Var("b1")]), condition=And(Geq(Var("a1"), Const(1)), Leq(Var("a1"), MinOverSet(set=MapOverSet(set=SolutionsSet(var=Tuple(el...
ALG
null
COMPUTE
sympy
SUM_GEOM
[ "B3_DIFF/POLY4_COUNT" ]
848bcb
algebra_quadratic_discriminant_v1
null
6
0
[ "B3_DIFF", "POLY4_COUNT", "SUM_GEOM" ]
3
0.225
2026-03-10T04:54:46.423497Z
{ "verified": true, "answer": 225, "timestamp": "2026-03-10T04:54:46.648266Z" }
a919d1
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 278, "completion_tokens": 1492 }, "timestamp": "2026-03-29T11:52:10.859Z", "answer": 225 }, { "id...
1
[ { "lemma": "B3_DIFF", "status": "ok" }, { "lemma": "POLY4_COUNT", "status": "ok_later" }, { "lemma": "POLY_PADIC_VAL_CONST", "status": "no" } ]
{ "lo": -6.2, "mid": -2.87, "hi": 0.44 }
4a7dc9
modular_count_residue_v1_1918700295_679
Let $m$ be the largest prime number in the interval $[2, 8]$. Determine the number of positive integers $n \leq 65025$ such that $n \equiv 1 \pmod{m}$.
9,290
graphs = [ Graph( let={ "_n": Const(2), "upper": Const(65025), "m": MaxOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), Ref("_n")), Leq(Var("n"), Const(8)), IsPrime(Var("n"))))), "r": Const(1), "result": CountOverSet(set=Solutions...
NT
null
COUNT
sympy
MAX_PRIME_BELOW
[ "MAX_PRIME_BELOW" ]
dc3ad3
modular_count_residue_v1
null
3
0
[ "MAX_PRIME_BELOW" ]
1
2.681
2026-02-08T03:22:50.841384Z
{ "verified": true, "answer": 9290, "timestamp": "2026-02-08T03:22:53.522615Z" }
13db61
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 161, "completion_tokens": 925 }, "timestamp": "2026-02-10T14:10:46.950Z", "answer": 9290 }, { "id...
1
[ { "lemma": "K5", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "LTE_SUM", "status": "no" }, { "lemma": "MAX_PRIME_BELOW", "status": "ok" }, { "lemma": "MOD_MUL", "status": "no" }, { "lemma": "V8_SUM", "status": "no" } ]
{ "lo": -7.01, "mid": -4.68, "hi": -2.15 }
d1d2eb
comb_catalan_compute_v1_458359167_1878
Let $m$ be the number of integers $t$ with $9 \leq t \leq 64$ such that there exist positive integers $a$, $b$ with $1 \leq a \leq 6$, $1 \leq b \leq 8$, and $t = 4a + 5b$. Let $n$ be the number of ordered pairs $(x_1, x_2)$ of positive odd integers such that $x_1 + x_2 = m$. Define $p$ to be the number of ordered pair...
58,786
graphs = [ Graph( let={ "_m": CountOverSet(set=SolutionsSet(var=Var("t"), condition=Exists(var=Var(name='a'), condition=Exists(var=Var(name='b'), condition=And(Geq(left=Var(name='a'), right=Const(value=1)), Leq(left=Var(name='a'), right=Const(value=6)), Geq(left=Var(name='b'), right=Const(value=...
COMB
null
COMPUTE
sympy
LIN_FORM
[ "LIN_FORM/COMB1", "COMB1/COMB1" ]
818a72
comb_catalan_compute_v1
null
7
0
[ "COMB1", "LIN_FORM" ]
2
0.005
2026-02-08T04:55:14.516335Z
{ "verified": true, "answer": 58786, "timestamp": "2026-02-08T04:55:14.521020Z" }
cab436
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 268, "completion_tokens": 6037 }, "timestamp": "2026-02-11T22:27:09.535Z", "answer": 58786 }, { "...
1
[ { "lemma": "C2", "status": "no" }, { "lemma": "COMB1", "status": "ok" }, { "lemma": "COUNT_CARTESIAN", "status": "no" }, { "lemma": "COUNT_INTEGER_RANGE", "status": "no" }, { "lemma": "LIN_FORM", "status": "ok" }, { "lemma": "V7", "status": "no" ...
{ "lo": -0.05, "mid": 2.89, "hi": 5.27 }
f1d50b
antilemma_k2_v1_153355830_2196
Let $n = 64$. Compute the value of $$ \sum_{k=1}^{d(n)} \varphi(k) \left\lfloor \frac{64}{k} \right\rfloor, $$ where $d(n) = \sum_{d \mid n} \varphi(d)$ and $\varphi$ denotes Euler's totient function.
2,080
graphs = [ Graph( let={ "_n": Const(64), "x": Summation(var="k", start=Const(1), end=SumOverDivisors(n=SumOverDivisors(n=Const(value=64), var='d', expr=EulerPhi(n=Var(name='d'))), var='d', expr=EulerPhi(n=Var(name='d'))), expr=Mul(EulerPhi(n=Var("k")), Floor(Div(Ref("_n"), Var("k")))...
NT
COMB
COMPUTE
sympy
K3
[ "K3/K3/K2", "K2" ]
d92398
antilemma_k2_v1
null
5
0
[ "K2", "K3" ]
2
0.001
2026-02-08T06:57:52.094241Z
{ "verified": true, "answer": 2080, "timestamp": "2026-02-08T06:57:52.095412Z" }
2ab932
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": 655 }, "timestamp": "2026-02-13T06:42:30.024Z", "answer": 2080 }, { ...
1
[ { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "DS2", "status": "no" }, { "lemma": "K18", "status": "no" }, { "lemma": "K2", "status": "ok" }, { "lemma": "K3", "status": "ok" }, { "lemma": "LTE_DIFF", "status": "no" }, { "le...
{ "lo": -3.51, "mid": 1.33, "hi": 6.56 }
eed70e
antilemma_k3_v1_1874849503_721
Let $x = \sum_{d \mid 82629} \phi(d)$. Compute the remainder when $44121x$ is divided by $68993$.
14,996
graphs = [ Graph( let={ "x": SumOverDivisors(n=Const(value=82629), var='d', expr=EulerPhi(n=Var(name='d'))), "Q": Mod(value=Mul(Const(44121), Ref("x")), modulus=Const(68993)), }, goal=Ref("Q"), ) ]
NT
COMB
COMPUTE
sympy
K3
[ "K3" ]
54c41e
antilemma_k3_v1
null
3
0
[ "K3" ]
1
0.001
2026-02-08T13:16:21.458533Z
{ "verified": true, "answer": 14996, "timestamp": "2026-02-08T13:16:21.459468Z" }
43b3a2
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 3, "correct": { "strict": true, "boxed": true, "relaxed": true }, "usage": { "prompt_tokens": 145, "completion_tokens": 6128 }, "timestamp": "2026-02-09T20:15:47.451Z", "answer": 14996 }, { "...
1
[ { "lemma": "COUNT_FIB_DIVISIBLE", "status": "no" }, { "lemma": "K13", "status": "no" }, { "lemma": "K3", "status": "ok" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "LTE_DIFF_P2", "status": "no" }, { "lemma": "V3", "status": "no" } ]
{ "lo": -5.49, "mid": 0.03, "hi": 6.12 }
cddfcb
nt_count_primes_v1_1456120455_9
Let $r$ be the sum of all real solutions to the equation $x^2 - 2x - 5928 = 0$. Let $T$ be the set of all prime numbers $n$ such that $r \leq n \leq 10609$. Compute the number of elements in $T$.
1,294
graphs = [ Graph( let={ "upper": Const(10609), "result": CountOverSet(set=SolutionsSet(var=Var("n"), condition=And(Geq(Var("n"), SumOverSet(set=SolutionsSet(var=Var("x"), condition=Eq(Sum(Pow(Var("x"), Const(2)), Mul(Const(-2), Var("x")), Const(-5928)), Const(0))))), Leq(Var("n"), Re...
NT
null
COUNT
sympy
VIETA_SUM
[ "VIETA_SUM" ]
b33a7a
nt_count_primes_v1
null
4
0
[ "VIETA_SUM" ]
1
0.936
2026-02-08T02:48:30.342688Z
{ "verified": true, "answer": 1294, "timestamp": "2026-02-08T02:48:31.278550Z" }
aa2b51
CC BY 4.0
[ { "id": 1, "model": "openai/gpt-oss-20b", "score": 0, "correct": { "strict": false, "boxed": false, "relaxed": false }, "usage": { "prompt_tokens": 178, "completion_tokens": 32768 }, "timestamp": "2026-02-23T17:42:53.492Z", "answer": null }, { ...
0
[ { "lemma": "COUNT_COPRIME_GRID", "status": "no" }, { "lemma": "K17", "status": "no" }, { "lemma": "K5", "status": "no" }, { "lemma": "MOD_ADD", "status": "no" }, { "lemma": "MOD_SUB", "status": "no" }, { "lemma": "VIETA_SUM", "status": "ok" } ]
{ "lo": 5.42, "mid": 7.53, "hi": 10 }
6dadeb
sequence_lucas_compute_v1_48377204_1572
Let $S$ be the set of all ordered pairs of positive integers $(x, y)$ such that $xy = 81$. Let $n$ be the minimum value of $x + y$ over all pairs $(x, y) \in S$. Compute the $n$-th Lucas number, and then find the remainder when $44121$ times this number is divided by $87581$.
70,428
graphs = [ Graph( let={ "_n": Const(44121), "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(81)))), expr=Sum(Var("x"), Var("y")))), ...
NT
null
COMPUTE
sympy
B3
[ "B3" ]
0cd20d
sequence_lucas_compute_v1
null
4
0
[ "B3" ]
1
0.002
2026-02-08T16:12:34.782397Z
{ "verified": true, "answer": 70428, "timestamp": "2026-02-08T16:12:34.784504Z" }
e39b8a
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": 1148 }, "timestamp": "2026-02-16T23:28:00.918Z", "answer": 70428 }, ...
1
[ { "lemma": "B3", "status": "ok" }, { "lemma": "COUNT_PRIMES", "status": "no" }, { "lemma": "LTE_DIFF", "status": "no" }, { "lemma": "V1", "status": "no" }, { "lemma": "V7", "status": "no" }, { "lemma": "VAL_SUM_EQ", "status": "no" } ]
{ "lo": -7.08, "mid": -0.29, "hi": 6.49 }