Datasets:
id string | problem string | answer string | k int64 | parts string | subdomain string | difficulty float64 | difficulty_mean float64 | combine string | grouping string | source string | origin string |
|---|---|---|---|---|---|---|---|---|---|---|---|
comp-train-00-00000 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
It is known that exactly one of the three (distinguishable) musketeers stole the truffles. Each musketeer makes one statement... | 104 | 3 | [{"id": "omni-01118", "subdomain": "Applied Mathematics / Statistics", "difficulty": 4.0, "answer": 99}, {"id": "omni-02619", "subdomain": "Applied Mathematics / Statistics", "difficulty": 5.25, "answer": 0}, {"id": "omni-01722", "subdomain": "Applied Mathematics / Statistics", "difficulty": 5.25, "answer": 5}] | Applied Mathematics / Statistics | 5.25 | 4.833333 | sum | subdomain | omni-math-composite | |
comp-train-00-00001 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Call a triangle nice if the plane can be tiled using congruent copies of this triangle so that any two triangles that share a... | 698 | 3 | [{"id": "omni-01912", "subdomain": "Geometry / Plane Geometry", "difficulty": 5.0, "answer": 4}, {"id": "omni-02542", "subdomain": "Geometry / Plane Geometry", "difficulty": 4.0, "answer": 280}, {"id": "omni-02126", "subdomain": "Geometry / Plane Geometry", "difficulty": 4.0, "answer": 414}] | Geometry / Plane Geometry | 5 | 4.333333 | sum | subdomain | omni-math-composite | |
comp-train-00-00002 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
How many times does the letter "e" occur in all problem statements in this year's HMMT February competition?
Part 2:
Conside... | 1665 | 2 | [{"id": "omni-01068", "subdomain": "Applied Mathematics / Math Word Problems", "difficulty": 5.25, "answer": 1661}, {"id": "omni-01750", "subdomain": "Applied Mathematics / Math Word Problems", "difficulty": 4.5, "answer": 4}] | Applied Mathematics / Math Word Problems | 5.25 | 4.875 | sum | subdomain | omni-math-composite | |
comp-train-00-00003 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Let $A$ be the area of the largest semicircle that can be inscribed in a quarter-circle of radius 1. Compute $\frac{120 A}{\p... | 36 | 2 | [{"id": "omni-02021", "subdomain": "Geometry / Plane Geometry", "difficulty": 4.5, "answer": 20}, {"id": "omni-02517", "subdomain": "Geometry / Plane Geometry", "difficulty": 4.0, "answer": 16}] | Geometry / Plane Geometry | 4.5 | 4.25 | sum | subdomain | omni-math-composite | |
comp-train-00-00004 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
How many ways are there to color the vertices of a triangle red, green, blue, or yellow such that no two vertices have the sa... | 1433 | 2 | [{"id": "omni-02043", "subdomain": "Applied Mathematics / Statistics", "difficulty": 4.0, "answer": 24}, {"id": "omni-02595", "subdomain": "Applied Mathematics / Statistics", "difficulty": 5.0, "answer": 1409}] | Applied Mathematics / Statistics | 5 | 4.5 | sum | subdomain | omni-math-composite | |
comp-train-00-00005 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Call a string of letters $S$ an almost palindrome if $S$ and the reverse of $S$ differ in exactly two places. Find the number... | 4891 | 2 | [{"id": "omni-01791", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.0, "answer": 2160}, {"id": "omni-01987", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.0, "answer": 2731}] | Discrete Mathematics / mixed | 5 | 5 | sum | domain | omni-math-composite | |
comp-train-00-00006 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Find the smallest possible value of $x+y$ where $x, y \geq 1$ and $x$ and $y$ are integers that satisfy $x^{2}-29y^{2}=1$
Pa... | 11701 | 3 | [{"id": "omni-00469", "subdomain": "Number Theory / Other", "difficulty": 5.0, "answer": 11621}, {"id": "omni-00553", "subdomain": "Number Theory / Other", "difficulty": 4.0, "answer": 20}, {"id": "omni-03368", "subdomain": "Number Theory / Other", "difficulty": 4.5, "answer": 60}] | Number Theory / Other | 5 | 4.5 | sum | subdomain | omni-math-composite | |
comp-train-00-00007 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
In an acute scalene triangle $ABC$, points $D,E,F$ lie on sides $BC, CA, AB$, respectively, such that $AD \perp BC, BE \perp ... | 33803 | 3 | [{"id": "omni-00001", "subdomain": "Geometry / Plane Geometry", "difficulty": 7.0, "answer": 1}, {"id": "omni-02578", "subdomain": "Geometry / Plane Geometry", "difficulty": 4.5, "answer": 33725}, {"id": "omni-02024", "subdomain": "Geometry / Plane Geometry", "difficulty": 4.5, "answer": 77}] | Geometry / Plane Geometry | 7 | 5.333333 | sum | subdomain | omni-math-composite | |
comp-train-00-00008 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Square \(ABCD\) is inscribed in circle \(\omega\) with radius 10. Four additional squares are drawn inside \(\omega\) but out... | 432 | 2 | [{"id": "omni-00475", "subdomain": "Geometry / Plane Geometry", "difficulty": 5.0, "answer": 144}, {"id": "omni-01338", "subdomain": "Geometry / Plane Geometry", "difficulty": 5.25, "answer": 288}] | Geometry / Plane Geometry | 5.25 | 5.125 | sum | subdomain | omni-math-composite | |
comp-train-00-00009 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Let $ S \equal{} \{1,2,3,\cdots ,280\}$. Find the smallest integer $ n$ such that each $ n$-element subset of $ S$ contains f... | 283 | 2 | [{"id": "omni-03844", "subdomain": "Number Theory / Prime Numbers", "difficulty": 6.5, "answer": 217}, {"id": "omni-02481", "subdomain": "Number Theory / Prime Numbers", "difficulty": 4.0, "answer": 66}] | Number Theory / Prime Numbers | 6.5 | 5.25 | sum | subdomain | omni-math-composite | |
comp-train-00-00010 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Find the number of digits in the decimal representation of $2^{41}$.
Part 2:
Let $f$ be a function that takes in a triple of... | 23 | 3 | [{"id": "omni-01840", "subdomain": "Precalculus / Functions", "difficulty": 4.0, "answer": 13}, {"id": "omni-01666", "subdomain": "Precalculus / Functions", "difficulty": 4.5, "answer": 8}, {"id": "omni-00421", "subdomain": "Precalculus / Functions", "difficulty": 5.0, "answer": 2}] | Precalculus / Functions | 5 | 4.5 | sum | subdomain | omni-math-composite | |
comp-train-00-00011 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Let $ABCDE$ be a convex pentagon such that $AB=AE=CD=1$, $\angle ABC=\angle DEA=90^\circ$ and $BC+DE=1$. Compute the area of ... | 2096 | 3 | [{"id": "omni-03753", "subdomain": "Geometry / Plane Geometry", "difficulty": 5.0, "answer": 1}, {"id": "omni-02024", "subdomain": "Geometry / Plane Geometry", "difficulty": 4.5, "answer": 77}, {"id": "omni-03721", "subdomain": "Geometry / Plane Geometry", "difficulty": 7.0, "answer": 2018}] | Geometry / Plane Geometry | 7 | 5.5 | sum | subdomain | omni-math-composite | |
comp-train-00-00012 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Point $P$ lies inside equilateral triangle $A B C$ so that $\angle B P C=120^{\circ}$ and $A P \sqrt{2}=B P+C P$. $\frac{A P}... | 2273 | 2 | [{"id": "omni-02591", "subdomain": "Geometry / Plane Geometry", "difficulty": 5.0, "answer": 255}, {"id": "omni-03721", "subdomain": "Geometry / Plane Geometry", "difficulty": 7.0, "answer": 2018}] | Geometry / mixed | 7 | 6 | sum | domain | omni-math-composite | |
comp-train-00-00013 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Consider a permutation $\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right)$ of $\{1,2,3,4,5\}$. We say the tuple $\left(a_{1}, a_... | 3426600 | 3 | [{"id": "omni-01777", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 4.5, "answer": 20}, {"id": "omni-01560", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.0, "answer": 4}, {"id": "omni-02456", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.25, "answer": ... | Discrete Mathematics / Combinatorics | 5.25 | 4.916667 | sum | subdomain | omni-math-composite | |
comp-train-00-00014 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
For each positive integer $n$ and non-negative integer $k$, define $W(n, k)$ recursively by $$ W(n, k)= \begin{cases}n^{n} & ... | 1212 | 2 | [{"id": "omni-01051", "subdomain": "Algebra / Intermediate Algebra", "difficulty": 5.0, "answer": 875}, {"id": "omni-00131", "subdomain": "Algebra / Intermediate Algebra", "difficulty": 7.0, "answer": 337}] | Algebra / Intermediate Algebra | 7 | 6 | sum | subdomain | omni-math-composite | |
comp-train-00-00015 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Given $\frac{e}{f}=\frac{3}{4}$ and $\sqrt{e^{2}+f^{2}}=15$, find $ef$.
Part 2:
Determine the number of triples $0 \leq k, m... | 131 | 3 | [{"id": "omni-00931", "subdomain": "Algebra / Algebra", "difficulty": 5.0, "answer": 108}, {"id": "omni-01101", "subdomain": "Algebra / Algebra", "difficulty": 5.5, "answer": 22}, {"id": "omni-00449", "subdomain": "Algebra / Algebra", "difficulty": 4.5, "answer": 1}] | Algebra / Algebra | 5.5 | 5 | sum | subdomain | omni-math-composite | |
comp-train-00-00016 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Find the number of positive integers less than 1000000 that are divisible by some perfect cube greater than 1.
Part 2:
A rep... | 1391545 | 2 | [{"id": "omni-02293", "subdomain": "Number Theory / Other", "difficulty": 5.25, "answer": 168089}, {"id": "omni-02581", "subdomain": "Number Theory / Other", "difficulty": 5.0, "answer": 1223456}] | Number Theory / Other | 5.25 | 5.125 | sum | subdomain | omni-math-composite | |
comp-train-00-00017 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
All positive integers whose binary representations (excluding leading zeroes) have at least as many 1's as 0's are put in inc... | 132184 | 3 | [{"id": "omni-02295", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.0, "answer": 9}, {"id": "omni-01263", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.0, "answer": 131922}, {"id": "omni-01177", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 8.0, "answer... | Discrete Mathematics / Combinatorics | 8 | 6 | sum | subdomain | omni-math-composite | |
comp-train-00-00018 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
A dance with 2018 couples takes place in Havana. For the dance, 2018 distinct points labeled $0, 1,\ldots, 2017$ are marked i... | 895 | 3 | [{"id": "omni-03608", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.5, "answer": 505}, {"id": "omni-03213", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 7.0, "answer": 290}, {"id": "omni-03871", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 9.0, "answer"... | Discrete Mathematics / Combinatorics | 9 | 7.166667 | sum | subdomain | omni-math-composite | |
comp-train-00-00019 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
The vertices of a regular nonagon are colored such that 1) adjacent vertices are different colors and 2) if 3 vertices form a... | 100138 | 3 | [{"id": "omni-02159", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 4.0, "answer": 54}, {"id": "omni-02510", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.25, "answer": 100000}, {"id": "omni-02652", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.0, "answ... | Discrete Mathematics / Combinatorics | 5.25 | 4.75 | sum | subdomain | omni-math-composite | |
comp-train-00-00020 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Let $a, b, c, x, y$, and $z$ be complex numbers such that $a=\frac{b+c}{x-2}, \quad b=\frac{c+a}{y-2}, \quad c=\frac{a+b}{z-2... | 59658 | 3 | [{"id": "omni-01388", "subdomain": "Algebra / Intermediate Algebra", "difficulty": 5.5, "answer": -5892}, {"id": "omni-00821", "subdomain": "Algebra / Intermediate Algebra", "difficulty": 5.0, "answer": 65536}, {"id": "omni-01245", "subdomain": "Algebra / Intermediate Algebra", "difficulty": 4.0, "answer": 14}] | Algebra / Intermediate Algebra | 5.5 | 4.833333 | sum | subdomain | omni-math-composite | |
comp-train-00-00021 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
A point $(x, y)$ is selected uniformly at random from the unit square $S=\{(x, y) \mid 0 \leq x \leq 1,0 \leq y \leq 1\}$. If... | 987 | 3 | [{"id": "omni-02468", "subdomain": "Applied Mathematics / Statistics", "difficulty": 5.0, "answer": 820}, {"id": "omni-00597", "subdomain": "Applied Mathematics / Statistics", "difficulty": 4.0, "answer": 24}, {"id": "omni-00779", "subdomain": "Applied Mathematics / Statistics", "difficulty": 5.0, "answer": 143}] | Applied Mathematics / Statistics | 5 | 4.666667 | sum | subdomain | omni-math-composite | |
comp-train-00-00022 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
The sequence $\{a_n\}_{n\geq 0}$ of real numbers satisfies the relation:
\[ a_{m+n} + a_{m-n} - m + n -1 = \frac12 (a_{2m} + ... | 4018026 | 2 | [{"id": "omni-03862", "subdomain": "Algebra / Intermediate Algebra", "difficulty": 5.0, "answer": 4018021}, {"id": "omni-02558", "subdomain": "Algebra / Intermediate Algebra", "difficulty": 5.0, "answer": 5}] | Algebra / Intermediate Algebra | 5 | 5 | sum | subdomain | omni-math-composite | |
comp-train-00-00023 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Ten points are equally spaced on a circle. A graph is a set of segments (possibly empty) drawn between pairs of points, so th... | 11716575 | 2 | [{"id": "omni-01795", "subdomain": "Discrete Mathematics / Graph Theory", "difficulty": 6.0, "answer": 11716571}, {"id": "omni-03593", "subdomain": "Discrete Mathematics / Graph Theory", "difficulty": 6.0, "answer": 4}] | Discrete Mathematics / Graph Theory | 6 | 6 | sum | subdomain | omni-math-composite | |
comp-train-00-00024 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
A string of digits is defined to be similar to another string of digits if it can be obtained by reversing some contiguous su... | 3147 | 2 | [{"id": "omni-02645", "subdomain": "Applied Mathematics / Statistics", "difficulty": 5.0, "answer": 1126}, {"id": "omni-00833", "subdomain": "Applied Mathematics / Statistics", "difficulty": 5.0, "answer": 2021}] | Applied Mathematics / Statistics | 5 | 5 | sum | subdomain | omni-math-composite | |
comp-train-00-00025 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
For each positive integer $n$ and non-negative integer $k$, define $W(n, k)$ recursively by $$ W(n, k)= \begin{cases}n^{n} & ... | 1307 | 2 | [{"id": "omni-01051", "subdomain": "Algebra / Intermediate Algebra", "difficulty": 5.0, "answer": 875}, {"id": "omni-00834", "subdomain": "Algebra / Intermediate Algebra", "difficulty": 4.5, "answer": 432}] | Algebra / Intermediate Algebra | 5 | 4.75 | sum | subdomain | omni-math-composite | |
comp-train-00-00026 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Tetrahedron $A B C D$ has side lengths $A B=6, B D=6 \sqrt{2}, B C=10, A C=8, C D=10$, and $A D=6$. The distance from vertex ... | 3340 | 3 | [{"id": "omni-02573", "subdomain": "Geometry / Solid Geometry", "difficulty": 5.0, "answer": 2851}, {"id": "omni-00046", "subdomain": "Geometry / Solid Geometry", "difficulty": 8.0, "answer": 9}, {"id": "omni-01890", "subdomain": "Geometry / Solid Geometry", "difficulty": 4.5, "answer": 480}] | Geometry / Solid Geometry | 8 | 5.833333 | sum | subdomain | omni-math-composite | |
comp-train-00-00027 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
At a recent math contest, Evan was asked to find $2^{2016}(\bmod p)$ for a given prime number $p$ with $100<p<500$. Evan has ... | 254 | 3 | [{"id": "omni-00460", "subdomain": "Number Theory / Congruences", "difficulty": 5.0, "answer": 211}, {"id": "omni-01031", "subdomain": "Number Theory / Congruences", "difficulty": 5.0, "answer": 12}, {"id": "omni-03811", "subdomain": "Number Theory / Congruences", "difficulty": 7.0, "answer": 31}] | Number Theory / Congruences | 7 | 5.666667 | sum | subdomain | omni-math-composite | |
comp-train-00-00028 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Mary has a sequence $m_{2}, m_{3}, m_{4}, \ldots$, such that for each $b \geq 2, m_{b}$ is the least positive integer $m$ for... | 2185 | 3 | [{"id": "omni-02430", "subdomain": "Algebra / Intermediate Algebra", "difficulty": 4.0, "answer": 2188}, {"id": "omni-02158", "subdomain": "Algebra / Intermediate Algebra", "difficulty": 4.5, "answer": 1}, {"id": "omni-00405", "subdomain": "Algebra / Intermediate Algebra", "difficulty": 4.0, "answer": -4}] | Algebra / Intermediate Algebra | 4.5 | 4.166667 | sum | subdomain | omni-math-composite | |
comp-train-00-00029 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
It is known that exactly one of the three (distinguishable) musketeers stole the truffles. Each musketeer makes one statement... | 7659 | 2 | [{"id": "omni-01118", "subdomain": "Applied Mathematics / Statistics", "difficulty": 4.0, "answer": 99}, {"id": "omni-01218", "subdomain": "Applied Mathematics / Statistics", "difficulty": 4.0, "answer": 7560}] | Applied Mathematics / Statistics | 4 | 4 | sum | subdomain | omni-math-composite | |
comp-train-00-00030 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Call a positive integer $N \geq 2$ "special" if for every $k$ such that $2 \leq k \leq N, N$ can be expressed as a sum of $k$... | 2454794 | 3 | [{"id": "omni-01321", "subdomain": "Number Theory / Other", "difficulty": 4.0, "answer": 50}, {"id": "omni-03324", "subdomain": "Number Theory / Other", "difficulty": 5.0, "answer": 1231288}, {"id": "omni-02581", "subdomain": "Number Theory / Other", "difficulty": 5.0, "answer": 1223456}] | Number Theory / Other | 5 | 4.666667 | sum | subdomain | omni-math-composite | |
comp-train-00-00031 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Determine the number of triples $0 \leq k, m, n \leq 100$ of integers such that $$ 2^{m} n-2^{n} m=2^{k} $$
Part 2:
Let $f: ... | 2237 | 2 | [{"id": "omni-01101", "subdomain": "Algebra / Algebra", "difficulty": 5.5, "answer": 22}, {"id": "omni-01145", "subdomain": "Algebra / Intermediate Algebra", "difficulty": 5.0, "answer": 2215}] | Algebra / mixed | 5.5 | 5.25 | sum | domain | omni-math-composite | |
comp-train-00-00032 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Compute the number of ordered pairs of integers $(x, y)$ such that $x^{2}+y^{2}<2019$ and $$x^{2}+\min (x, y)=y^{2}+\max (x, ... | 148 | 3 | [{"id": "omni-02210", "subdomain": "Algebra / Algebra", "difficulty": 5.0, "answer": 127}, {"id": "omni-01249", "subdomain": "Algebra / Algebra", "difficulty": 4.5, "answer": 8}, {"id": "omni-00381", "subdomain": "Algebra / Algebra", "difficulty": 5.0, "answer": 13}] | Algebra / Algebra | 5 | 4.833333 | sum | subdomain | omni-math-composite | |
comp-train-00-00033 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
How many positive integers less than 100 are relatively prime to 200?
Part 2:
Find the smallest positive integer $n$ such th... | 737 | 3 | [{"id": "omni-02290", "subdomain": "Number Theory / Greatest Common Divisors (GCD)", "difficulty": 4.0, "answer": 40}, {"id": "omni-03639", "subdomain": "Number Theory / Greatest Common Divisors (GCD)", "difficulty": 5.0, "answer": 95}, {"id": "omni-01308", "subdomain": "Number Theory / Greatest Common Divisors (GCD)",... | Number Theory / Greatest Common Divisors (GCD) | 5 | 4.333333 | sum | subdomain | omni-math-composite | |
comp-train-00-00034 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
FIx positive integer $n$. Prove: For any positive integers $a,b,c$ not exceeding $3n^2+4n$, there exist integers $x,y,z$ with... | 5 | 3 | [{"id": "omni-00303", "subdomain": "Algebra / Linear Algebra", "difficulty": 9.0, "answer": 0}, {"id": "omni-02640", "subdomain": "Algebra / Linear Algebra", "difficulty": 7.0, "answer": 3}, {"id": "omni-03759", "subdomain": "Algebra / Linear Algebra", "difficulty": 7.0, "answer": 2}] | Algebra / Linear Algebra | 9 | 7.666667 | sum | subdomain | omni-math-composite | |
comp-train-00-00035 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Compute the number of labelings $f:\{0,1\}^{3} \rightarrow\{0,1, \ldots, 7\}$ of the vertices of the unit cube such that $$\l... | 967 | 3 | [{"id": "omni-00941", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.25, "answer": 144}, {"id": "omni-01277", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.0, "answer": 43}, {"id": "omni-02135", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.0, "answer"... | Discrete Mathematics / Combinatorics | 5.25 | 5.083333 | sum | subdomain | omni-math-composite | |
comp-train-00-00036 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Circles $C_{1}, C_{2}, C_{3}$ have radius 1 and centers $O, P, Q$ respectively. $C_{1}$ and $C_{2}$ intersect at $A, C_{2}$ a... | 3210 | 3 | [{"id": "omni-00960", "subdomain": "Geometry / Plane Geometry", "difficulty": 5.0, "answer": 90}, {"id": "omni-01936", "subdomain": "Geometry / Plane Geometry", "difficulty": 4.0, "answer": 1}, {"id": "omni-02222", "subdomain": "Geometry / Plane Geometry", "difficulty": 4.0, "answer": 3119}] | Geometry / Plane Geometry | 5 | 4.333333 | sum | subdomain | omni-math-composite | |
comp-train-00-00037 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of
$$\frac{a... | 2031 | 3 | [{"id": "omni-03879", "subdomain": "Algebra / Algebra", "difficulty": 7.0, "answer": 8}, {"id": "omni-01296", "subdomain": "Algebra / Algebra", "difficulty": 4.0, "answer": 2002}, {"id": "omni-01689", "subdomain": "Algebra / Algebra", "difficulty": 4.0, "answer": 21}] | Algebra / Algebra | 7 | 5 | sum | subdomain | omni-math-composite | |
comp-train-00-00038 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
There are 5 students on a team for a math competition. The math competition has 5 subject tests. Each student on the team mus... | 2042 | 2 | [{"id": "omni-01052", "subdomain": "Discrete Mathematics / Graph Theory", "difficulty": 4.0, "answer": 2040}, {"id": "omni-03587", "subdomain": "Discrete Mathematics / Graph Theory", "difficulty": 7.0, "answer": 2}] | Discrete Mathematics / Graph Theory | 7 | 5.5 | sum | subdomain | omni-math-composite | |
comp-train-00-00039 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Determine the maximum integer $ n $ such that for each positive integer $ k \le \frac{n}{2} $ there are two positive divisors... | 247611 | 3 | [{"id": "omni-03666", "subdomain": "Number Theory / Factorization", "difficulty": 6.0, "answer": 24}, {"id": "omni-01956", "subdomain": "Number Theory / Factorization", "difficulty": 4.0, "answer": 247548}, {"id": "omni-04052", "subdomain": "Number Theory / Factorization", "difficulty": 7.0, "answer": 39}] | Number Theory / Factorization | 7 | 5.666667 | sum | subdomain | omni-math-composite | |
comp-train-00-00040 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Let $a, b$, and $c$ be the 3 roots of $x^{3}-x+1=0$. Find $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}$.
Part 2:
Find the sum ... | -3195 | 2 | [{"id": "omni-01206", "subdomain": "Algebra / Algebra", "difficulty": 5.5, "answer": -2}, {"id": "omni-02215", "subdomain": "Algebra / Algebra", "difficulty": 4.5, "answer": -3193}] | Algebra / Algebra | 5.5 | 5 | sum | subdomain | omni-math-composite | |
comp-train-00-00041 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
In an acute scalene triangle $ABC$, points $D,E,F$ lie on sides $BC, CA, AB$, respectively, such that $AD \perp BC, BE \perp ... | 2007 | 3 | [{"id": "omni-00001", "subdomain": "Geometry / Plane Geometry", "difficulty": 7.0, "answer": 1}, {"id": "omni-00238", "subdomain": "Geometry / Plane Geometry", "difficulty": 7.0, "answer": 1999}, {"id": "omni-02462", "subdomain": "Geometry / Plane Geometry", "difficulty": 4.0, "answer": 7}] | Geometry / mixed | 7 | 6 | sum | domain | omni-math-composite | |
comp-train-00-00042 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
We want to design a new chess piece, the American, with the property that (i) the American can never attack itself, and (ii) ... | 1514 | 3 | [{"id": "omni-01895", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.25, "answer": 1024}, {"id": "omni-01115", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 4.5, "answer": 233}, {"id": "omni-02384", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 6.0, "answe... | Discrete Mathematics / Combinatorics | 6 | 5.25 | sum | subdomain | omni-math-composite | |
comp-train-00-00043 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Find the largest prime factor of $-x^{10}-x^{8}-x^{6}-x^{4}-x^{2}-1$, where $x=2 i$, $i=\sqrt{-1}$.
Part 2:
Determine the po... | 17 | 2 | [{"id": "omni-00381", "subdomain": "Algebra / Algebra", "difficulty": 5.0, "answer": 13}, {"id": "omni-02051", "subdomain": "Algebra / Algebra", "difficulty": 5.0, "answer": 4}] | Algebra / Algebra | 5 | 5 | sum | subdomain | omni-math-composite | |
comp-train-00-00044 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Consider pairs $(f,g)$ of functions from the set of nonnegative integers to itself such that
[list]
[*]$f(0) \geq f(1) \geq ... | 477849 | 3 | [{"id": "omni-00020", "subdomain": "Algebra / Intermediate Algebra", "difficulty": 8.0, "answer": 115440}, {"id": "omni-02554", "subdomain": "Algebra / Intermediate Algebra", "difficulty": 5.0, "answer": 360864}, {"id": "omni-00657", "subdomain": "Algebra / Intermediate Algebra", "difficulty": 5.25, "answer": 1545}] | Algebra / Intermediate Algebra | 8 | 6.083333 | sum | subdomain | omni-math-composite | |
comp-train-00-00045 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Call a triangle nice if the plane can be tiled using congruent copies of this triangle so that any two triangles that share a... | 6 | 2 | [{"id": "omni-01912", "subdomain": "Geometry / Plane Geometry", "difficulty": 5.0, "answer": 4}, {"id": "omni-01770", "subdomain": "Geometry / Plane Geometry", "difficulty": 4.0, "answer": 2}] | Geometry / Plane Geometry | 5 | 4.5 | sum | subdomain | omni-math-composite | |
comp-train-00-00046 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Find the smallest positive integer $n$ for which $$1!2!\cdots(n-1)!>n!^{2}$$
Part 2:
In how many ways can we fill the cells ... | 584 | 2 | [{"id": "omni-01884", "subdomain": "Number Theory / Factorization", "difficulty": 4.5, "answer": 8}, {"id": "omni-03999", "subdomain": "Number Theory / Factorization", "difficulty": 5.0, "answer": 576}] | Number Theory / Factorization | 5 | 4.75 | sum | subdomain | omni-math-composite | |
comp-train-00-00047 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
For each positive integer $n$, let $s(n)$ be the sum of the squares of the digits of $n$. For example, $s(15)=1^2+5^2=26$. De... | 8 | 2 | [{"id": "omni-03722", "subdomain": "Number Theory / Other", "difficulty": 5.0, "answer": 1}, {"id": "omni-00819", "subdomain": "Number Theory / Other", "difficulty": 4.0, "answer": 7}] | Number Theory / Other | 5 | 4.5 | sum | subdomain | omni-math-composite | |
comp-train-00-00048 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Count the number of functions $f: \mathbb{Z} \rightarrow\{$ 'green', 'blue' $\}$ such that $f(x)=f(x+22)$ for all integers $x... | 39861 | 3 | [{"id": "omni-03342", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.0, "answer": 39601}, {"id": "omni-03306", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 4.5, "answer": 132}, {"id": "omni-01634", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 4.0, "answe... | Discrete Mathematics / Combinatorics | 5 | 4.5 | sum | subdomain | omni-math-composite | |
comp-train-00-00049 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
All the sequences consisting of five letters from the set $\{T, U, R, N, I, P\}$ (with repetitions allowed) are arranged in a... | 99 | 2 | [{"id": "omni-00430", "subdomain": "Algebra / Prealgebra", "difficulty": 4.0, "answer": 0}, {"id": "omni-00773", "subdomain": "Algebra / Prealgebra", "difficulty": 5.0, "answer": 99}] | Algebra / Prealgebra | 5 | 4.5 | sum | subdomain | omni-math-composite | |
comp-train-00-00050 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
On the blackboard, Amy writes 2017 in base-$a$ to get $133201_{a}$. Betsy notices she can erase a digit from Amy's number and... | 23 | 2 | [{"id": "omni-01958", "subdomain": "Algebra / Other", "difficulty": 4.0, "answer": 22}, {"id": "omni-04041", "subdomain": "Algebra / Other", "difficulty": 9.5, "answer": 1}] | Algebra / Other | 9.5 | 6.75 | sum | subdomain | omni-math-composite | |
comp-train-00-00051 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Let $f(n)$ be the number of distinct prime divisors of $n$ less than 6. Compute $$\sum_{n=1}^{2020} f(n)^{2}$$
Part 2:
Find ... | 4377 | 2 | [{"id": "omni-02532", "subdomain": "Number Theory / Prime Numbers", "difficulty": 5.0, "answer": 3431}, {"id": "omni-03556", "subdomain": "Number Theory / Prime Numbers", "difficulty": 6.0, "answer": 946}] | Number Theory / Prime Numbers | 6 | 5.5 | sum | subdomain | omni-math-composite | |
comp-train-00-00052 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
For each nonnegative integer $n$ we define $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$. Find the greatest common divisor of the numbers ... | 3633 | 3 | [{"id": "omni-03692", "subdomain": "Number Theory / Greatest Common Divisors (GCD)", "difficulty": 5.0, "answer": 7}, {"id": "omni-01308", "subdomain": "Number Theory / Greatest Common Divisors (GCD)", "difficulty": 4.0, "answer": 602}, {"id": "omni-03917", "subdomain": "Number Theory / Greatest Common Divisors (GCD)",... | Number Theory / Greatest Common Divisors (GCD) | 9 | 6 | sum | subdomain | omni-math-composite | |
comp-train-00-00053 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Find the largest positive integer $m$ which makes it possible to color several cells of a $70\times 70$ table red such that [... | 761 | 2 | [{"id": "omni-00090", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 8.0, "answer": 32}, {"id": "omni-00966", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.0, "answer": 729}] | Discrete Mathematics / Combinatorics | 8 | 6.5 | sum | subdomain | omni-math-composite | |
comp-train-00-00054 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Teresa the bunny has a fair 8-sided die. Seven of its sides have fixed labels $1,2, \ldots, 7$, and the label on the eighth s... | 108 | 2 | [{"id": "omni-01268", "subdomain": "Applied Mathematics / Statistics", "difficulty": 4.0, "answer": 104}, {"id": "omni-02228", "subdomain": "Applied Mathematics / Statistics", "difficulty": 4.0, "answer": 4}] | Applied Mathematics / Statistics | 4 | 4 | sum | subdomain | omni-math-composite | |
comp-train-00-00055 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
A quagga is an extinct chess piece whose move is like a knight's, but much longer: it can move 6 squares in any direction (up... | 429 | 2 | [{"id": "omni-00453", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.25, "answer": 68}, {"id": "omni-01377", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.0, "answer": 361}] | Discrete Mathematics / Combinatorics | 5.25 | 5.125 | sum | subdomain | omni-math-composite | |
comp-train-00-00056 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
A bug is on a corner of a cube. A healthy path for the bug is a path along the edges of the cube that starts and ends where t... | 40 | 2 | [{"id": "omni-01647", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.0, "answer": 6}, {"id": "omni-01833", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 7.0, "answer": 34}] | Discrete Mathematics / Combinatorics | 7 | 6 | sum | subdomain | omni-math-composite | |
comp-train-00-00057 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
A lame king is a chess piece that can move from a cell to any cell that shares at least one vertex with it, except for the ce... | 57 | 2 | [{"id": "omni-01277", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.0, "answer": 43}, {"id": "omni-00503", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.5, "answer": 14}] | Discrete Mathematics / Combinatorics | 5.5 | 5.25 | sum | subdomain | omni-math-composite | |
comp-train-00-00058 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
We have an $n$-gon, and each of its vertices is labeled with a number from the set $\{1, \ldots, 10\}$. We know that for any ... | 552 | 3 | [{"id": "omni-00674", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.25, "answer": 50}, {"id": "omni-01482", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.0, "answer": 376}, {"id": "omni-01850", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.25, "answer... | Discrete Mathematics / Combinatorics | 5.25 | 5.166667 | sum | subdomain | omni-math-composite | |
comp-train-00-00059 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
A sequence of real numbers $a_0, a_1, . . .$ is said to be good if the following three conditions hold.
(i) The value of $a_0... | 30624 | 3 | [{"id": "omni-03617", "subdomain": "Algebra / Algebra", "difficulty": 7.0, "answer": 60}, {"id": "omni-02354", "subdomain": "Algebra / Algebra", "difficulty": 5.0, "answer": 764}, {"id": "omni-02316", "subdomain": "Algebra / Algebra", "difficulty": 5.0, "answer": 29800}] | Algebra / Algebra | 7 | 5.666667 | sum | subdomain | omni-math-composite | |
comp-train-00-00060 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
What is the smallest positive integer $n$ which cannot be written in any of the following forms? - $n=1+2+\cdots+k$ for a pos... | 71 | 3 | [{"id": "omni-01721", "subdomain": "Number Theory / Prime Numbers", "difficulty": 5.0, "answer": 40}, {"id": "omni-00817", "subdomain": "Number Theory / Prime Numbers", "difficulty": 5.0, "answer": 26}, {"id": "omni-02161", "subdomain": "Number Theory / Prime Numbers", "difficulty": 4.0, "answer": 5}] | Number Theory / Prime Numbers | 5 | 4.666667 | sum | subdomain | omni-math-composite | |
comp-train-00-00061 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Compute the number of triples $(f, g, h)$ of permutations on $\{1,2,3,4,5\}$ such that $$ \begin{aligned} & f(g(h(x)))=h(g(f(... | 187 | 2 | [{"id": "omni-01151", "subdomain": "Applied Mathematics / Statistics", "difficulty": 4.5, "answer": 146}, {"id": "omni-01190", "subdomain": "Applied Mathematics / Statistics", "difficulty": 4.0, "answer": 41}] | Applied Mathematics / Statistics | 4.5 | 4.25 | sum | subdomain | omni-math-composite | |
comp-train-00-00062 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Let $p, q, r$ be primes such that $2 p+3 q=6 r$. Find $p+q+r$.
Part 2:
Let $n$ be the second smallest integer that can be wr... | 4111 | 2 | [{"id": "omni-02496", "subdomain": "Number Theory / Prime Numbers", "difficulty": 4.0, "answer": 7}, {"id": "omni-01767", "subdomain": "Number Theory / Factorization", "difficulty": 4.5, "answer": 4104}] | Number Theory / mixed | 4.5 | 4.25 | sum | domain | omni-math-composite | |
comp-train-00-00063 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Let $x$ and $y$ be positive real numbers and $\theta$ an angle such that $\theta \neq \frac{\pi}{2} n$ for any integer $n$. S... | 1283 | 2 | [{"id": "omni-01344", "subdomain": "Algebra / Algebra", "difficulty": 5.0, "answer": 4}, {"id": "omni-00571", "subdomain": "Algebra / Algebra", "difficulty": 5.0, "answer": 1279}] | Algebra / Algebra | 5 | 5 | sum | subdomain | omni-math-composite | |
comp-train-00-00064 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Mario is once again on a quest to save Princess Peach. Mario enters Peach's castle and finds himself in a room with 4 doors. ... | 6090 | 3 | [{"id": "omni-01727", "subdomain": "Applied Mathematics / Statistics", "difficulty": 5.0, "answer": 5460}, {"id": "omni-01135", "subdomain": "Applied Mathematics / Statistics", "difficulty": 6.0, "answer": 114}, {"id": "omni-01575", "subdomain": "Applied Mathematics / Statistics", "difficulty": 5.0, "answer": 516}] | Applied Mathematics / Statistics | 6 | 5.333333 | sum | subdomain | omni-math-composite | |
comp-train-00-00065 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
A regular dodecagon $P_{1} P_{2} \cdots P_{12}$ is inscribed in a unit circle with center $O$. Let $X$ be the intersection of... | 101 | 3 | [{"id": "omni-01427", "subdomain": "Geometry / Plane Geometry", "difficulty": 4.5, "answer": 45}, {"id": "omni-02313", "subdomain": "Geometry / Plane Geometry", "difficulty": 4.0, "answer": 49}, {"id": "omni-00608", "subdomain": "Geometry / Plane Geometry", "difficulty": 4.5, "answer": 7}] | Geometry / Plane Geometry | 4.5 | 4.333333 | sum | subdomain | omni-math-composite | |
comp-train-00-00066 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Find the number of digits in the decimal representation of $2^{41}$.
Part 2:
Let $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be a... | 267 | 3 | [{"id": "omni-01840", "subdomain": "Precalculus / Functions", "difficulty": 4.0, "answer": 13}, {"id": "omni-00837", "subdomain": "Precalculus / Functions", "difficulty": 4.0, "answer": 246}, {"id": "omni-01666", "subdomain": "Precalculus / Functions", "difficulty": 4.5, "answer": 8}] | Precalculus / Functions | 4.5 | 4.166667 | sum | subdomain | omni-math-composite | |
comp-train-00-00067 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
The sequence $a_{1}, a_{2}, a_{3}, \ldots$ of real numbers satisfies the recurrence $a_{n+1}=\frac{a_{n}^{2}-a_{n-1}+2 a_{n}}... | 117652 | 2 | [{"id": "omni-00437", "subdomain": "Algebra / Algebra", "difficulty": 5.0, "answer": 3}, {"id": "omni-02543", "subdomain": "Algebra / Algebra", "difficulty": 4.0, "answer": 117649}] | Algebra / Algebra | 5 | 4.5 | sum | subdomain | omni-math-composite | |
comp-train-00-00068 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Vijay chooses three distinct integers \(a, b, c\) from the set \(\{1,2,3,4,5,6,7,8,9,10,11\}\). If \(k\) is the minimum value... | 947 | 3 | [{"id": "omni-02393", "subdomain": "Algebra / Intermediate Algebra", "difficulty": 4.5, "answer": 990}, {"id": "omni-01749", "subdomain": "Algebra / Intermediate Algebra", "difficulty": 5.25, "answer": 47}, {"id": "omni-01333", "subdomain": "Algebra / Intermediate Algebra", "difficulty": 6.0, "answer": -90}] | Algebra / Intermediate Algebra | 6 | 5.25 | sum | subdomain | omni-math-composite | |
comp-train-00-00069 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Define the function $f: \mathbb{R} \rightarrow \mathbb{R}$ by $$f(x)= \begin{cases}\frac{1}{x^{2}+\sqrt{x^{4}+2 x}} & \text {... | 1178 | 2 | [{"id": "omni-02653", "subdomain": "Precalculus / Functions", "difficulty": 6.0, "answer": 932}, {"id": "omni-00837", "subdomain": "Precalculus / Functions", "difficulty": 4.0, "answer": 246}] | Precalculus / Functions | 6 | 5 | sum | subdomain | omni-math-composite | |
comp-train-00-00070 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
A bug is on a corner of a cube. A healthy path for the bug is a path along the edges of the cube that starts and ends where t... | 14 | 2 | [{"id": "omni-01647", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.0, "answer": 6}, {"id": "omni-02382", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 4.0, "answer": 8}] | Discrete Mathematics / Combinatorics | 5 | 4.5 | sum | subdomain | omni-math-composite | |
comp-train-00-00071 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
The vertices of a regular nonagon are colored such that 1) adjacent vertices are different colors and 2) if 3 vertices form a... | 5238 | 2 | [{"id": "omni-02159", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 4.0, "answer": 54}, {"id": "omni-00999", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.0, "answer": 5184}] | Discrete Mathematics / Combinatorics | 5 | 4.5 | sum | subdomain | omni-math-composite | |
comp-train-00-00072 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Mario is once again on a quest to save Princess Peach. Mario enters Peach's castle and finds himself in a room with 4 doors. ... | 19867 | 3 | [{"id": "omni-01727", "subdomain": "Applied Mathematics / Statistics", "difficulty": 5.0, "answer": 5460}, {"id": "omni-01681", "subdomain": "Applied Mathematics / Statistics", "difficulty": 5.25, "answer": 14400}, {"id": "omni-02119", "subdomain": "Applied Mathematics / Statistics", "difficulty": 4.0, "answer": 7}] | Applied Mathematics / Statistics | 5.25 | 4.75 | sum | subdomain | omni-math-composite | |
comp-train-00-00073 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Let $f(x)=x^{2}+6 x+7$. Determine the smallest possible value of $f(f(f(f(x))))$ over all real numbers $x$.
Part 2:
Define t... | 1201 | 3 | [{"id": "omni-02081", "subdomain": "Precalculus / Functions", "difficulty": 5.0, "answer": 23}, {"id": "omni-02653", "subdomain": "Precalculus / Functions", "difficulty": 6.0, "answer": 932}, {"id": "omni-00837", "subdomain": "Precalculus / Functions", "difficulty": 4.0, "answer": 246}] | Precalculus / Functions | 6 | 5 | sum | subdomain | omni-math-composite | |
comp-train-00-00074 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Let $A B C D E F$ be a convex hexagon with the following properties. (a) $\overline{A C}$ and $\overline{A E}$ trisect $\angl... | 7308 | 2 | [{"id": "omni-01980", "subdomain": "Geometry / Plane Geometry", "difficulty": 5.0, "answer": 7295}, {"id": "omni-02004", "subdomain": "Geometry / Plane Geometry", "difficulty": 5.25, "answer": 13}] | Geometry / Plane Geometry | 5.25 | 5.125 | sum | subdomain | omni-math-composite | |
comp-train-00-00075 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
A number is called [i]Norwegian[/i] if it has three distinct positive divisors whose sum is equal to $2022$. Determine the sm... | 3864 | 2 | [{"id": "omni-03828", "subdomain": "Number Theory / Factorization", "difficulty": 6.5, "answer": 1344}, {"id": "omni-01214", "subdomain": "Number Theory / Factorization", "difficulty": 5.0, "answer": 2520}] | Number Theory / Factorization | 6.5 | 5.75 | sum | subdomain | omni-math-composite | |
comp-train-00-00076 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Let \(ABC\) be a triangle with \(AB=8, AC=12\), and \(BC=5\). Let \(M\) be the second intersection of the internal angle bise... | 167 | 3 | [{"id": "omni-02517", "subdomain": "Geometry / Plane Geometry", "difficulty": 4.0, "answer": 16}, {"id": "omni-00475", "subdomain": "Geometry / Plane Geometry", "difficulty": 5.0, "answer": 144}, {"id": "omni-00608", "subdomain": "Geometry / Plane Geometry", "difficulty": 4.5, "answer": 7}] | Geometry / Plane Geometry | 5 | 4.5 | sum | subdomain | omni-math-composite | |
comp-train-00-00077 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Suppose $m>n>1$ are positive integers such that there exist $n$ complex numbers $x_{1}, x_{2}, \ldots, x_{n}$ for which - $x_... | 47 | 2 | [{"id": "omni-01515", "subdomain": "Algebra / Algebra", "difficulty": 5.25, "answer": 34}, {"id": "omni-00381", "subdomain": "Algebra / Algebra", "difficulty": 5.0, "answer": 13}] | Algebra / Algebra | 5.25 | 5.125 | sum | subdomain | omni-math-composite | |
comp-train-00-00078 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Determine the number of triples $0 \leq k, m, n \leq 100$ of integers such that $$ 2^{m} n-2^{n} m=2^{k} $$
Part 2:
Let $f(x... | 4391 | 3 | [{"id": "omni-01101", "subdomain": "Algebra / Algebra", "difficulty": 5.5, "answer": 22}, {"id": "omni-02268", "subdomain": "Algebra / Algebra", "difficulty": 5.0, "answer": 3989}, {"id": "omni-01930", "subdomain": "Algebra / Algebra", "difficulty": 5.0, "answer": 380}] | Algebra / Algebra | 5.5 | 5.166667 | sum | subdomain | omni-math-composite | |
comp-train-00-00079 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Let $S$ be the set of all points in the plane whose coordinates are positive integers less than or equal to 100 (so $S$ has $... | 5439 | 3 | [{"id": "omni-00225", "subdomain": "Geometry / Plane Geometry", "difficulty": 5.0, "answer": 4950}, {"id": "omni-02625", "subdomain": "Geometry / Plane Geometry", "difficulty": 5.25, "answer": 320}, {"id": "omni-00545", "subdomain": "Geometry / Plane Geometry", "difficulty": 4.5, "answer": 169}] | Geometry / mixed | 5.25 | 4.916667 | sum | domain | omni-math-composite | |
comp-train-00-00080 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Let $n$ be a three-digit integer with nonzero digits, not all of which are the same. Define $f(n)$ to be the greatest common ... | 6000 | 2 | [{"id": "omni-01779", "subdomain": "Number Theory / Greatest Common Divisors (GCD)", "difficulty": 4.0, "answer": 5994}, {"id": "omni-01232", "subdomain": "Number Theory / Greatest Common Divisors (GCD)", "difficulty": 4.0, "answer": 6}] | Number Theory / Greatest Common Divisors (GCD) | 4 | 4 | sum | subdomain | omni-math-composite | |
comp-train-00-00081 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Call a string of letters $S$ an almost palindrome if $S$ and the reverse of $S$ differ in exactly two places. Find the number... | 103024 | 3 | [{"id": "omni-01791", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.0, "answer": 2160}, {"id": "omni-02553", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.0, "answer": 864}, {"id": "omni-02510", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.25, "answe... | Discrete Mathematics / Combinatorics | 5.25 | 5.083333 | sum | subdomain | omni-math-composite | |
comp-train-00-00082 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Tetrahedron $A B C D$ has side lengths $A B=6, B D=6 \sqrt{2}, B C=10, A C=8, C D=10$, and $A D=6$. The distance from vertex ... | 2860 | 2 | [{"id": "omni-02573", "subdomain": "Geometry / Solid Geometry", "difficulty": 5.0, "answer": 2851}, {"id": "omni-00046", "subdomain": "Geometry / Solid Geometry", "difficulty": 8.0, "answer": 9}] | Geometry / Solid Geometry | 8 | 6.5 | sum | subdomain | omni-math-composite | |
comp-train-00-00083 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
A $10 \times 10$ table consists of 100 unit cells. A block is a $2 \times 2$ square consisting of 4 unit cells of the table. ... | 105 | 2 | [{"id": "omni-01614", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.5, "answer": 39}, {"id": "omni-00663", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 5.0, "answer": 66}] | Discrete Mathematics / Combinatorics | 5.5 | 5.25 | sum | subdomain | omni-math-composite | |
comp-train-00-00084 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Compute the sum of all positive integers $n$ such that $50 \leq n \leq 100$ and $2 n+3$ does not divide $2^{n!}-1$.
Part 2:
... | 306 | 3 | [{"id": "omni-01638", "subdomain": "Number Theory / Prime Numbers", "difficulty": 5.5, "answer": 222}, {"id": "omni-03813", "subdomain": "Number Theory / Prime Numbers", "difficulty": 8.0, "answer": 6}, {"id": "omni-02040", "subdomain": "Number Theory / Prime Numbers", "difficulty": 4.0, "answer": 78}] | Number Theory / Prime Numbers | 8 | 5.833333 | sum | subdomain | omni-math-composite | |
comp-train-00-00085 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Let $r=H_{1}$ be the answer to this problem. Given that $r$ is a nonzero real number, what is the value of $r^{4}+4 r^{3}+6 r... | 1053 | 3 | [{"id": "omni-00924", "subdomain": "Algebra / Algebra", "difficulty": 5.0, "answer": -1}, {"id": "omni-01419", "subdomain": "Algebra / Algebra", "difficulty": 4.5, "answer": 84}, {"id": "omni-01488", "subdomain": "Algebra / Algebra", "difficulty": 8.0, "answer": 970}] | Algebra / Algebra | 8 | 5.833333 | sum | subdomain | omni-math-composite | |
comp-train-00-00086 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
A polynomial $P$ with integer coefficients is called tricky if it has 4 as a root. A polynomial is called $k$-tiny if it has ... | 64914394 | 3 | [{"id": "omni-02130", "subdomain": "Algebra / Algebra", "difficulty": 5.5, "answer": 64912347}, {"id": "omni-00984", "subdomain": "Algebra / Algebra", "difficulty": 5.0, "answer": 10}, {"id": "omni-02398", "subdomain": "Algebra / Algebra", "difficulty": 4.5, "answer": 2037}] | Algebra / Algebra | 5.5 | 5 | sum | subdomain | omni-math-composite | |
comp-train-00-00087 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
An HMMT party has $m$ MIT students and $h$ Harvard students for some positive integers $m$ and $h$, For every pair of people ... | 355 | 2 | [{"id": "omni-01797", "subdomain": "Algebra / Algebra", "difficulty": 5.0, "answer": 342}, {"id": "omni-02121", "subdomain": "Algebra / Algebra", "difficulty": 4.0, "answer": 13}] | Algebra / Algebra | 5 | 4.5 | sum | subdomain | omni-math-composite | |
comp-train-00-00088 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Rahul has ten cards face-down, which consist of five distinct pairs of matching cards. During each move of his game, Rahul ch... | 147 | 2 | [{"id": "omni-00588", "subdomain": "Applied Mathematics / Statistics", "difficulty": 4.0, "answer": 4}, {"id": "omni-00779", "subdomain": "Applied Mathematics / Statistics", "difficulty": 5.0, "answer": 143}] | Applied Mathematics / Statistics | 5 | 4.5 | sum | subdomain | omni-math-composite | |
comp-train-00-00089 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
What is the smallest positive integer $t$ such that there exist integers $x_1,x_2,\ldots,x_t$ with \[x^3_1+x^3_2+\,\ldots\,+... | 2070 | 3 | [{"id": "omni-03982", "subdomain": "Algebra / Algebra", "difficulty": 9.0, "answer": 4}, {"id": "omni-04412", "subdomain": "Algebra / Algebra", "difficulty": 5.5, "answer": 1958}, {"id": "omni-00931", "subdomain": "Algebra / Algebra", "difficulty": 5.0, "answer": 108}] | Algebra / Algebra | 9 | 6.5 | sum | subdomain | omni-math-composite | |
comp-train-00-00090 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Let $S$ be the smallest subset of the integers with the property that $0 \in S$ and for any $x \in S$, we have $3 x \in S$ an... | 204 | 3 | [{"id": "omni-01634", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 4.0, "answer": 128}, {"id": "omni-01127", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 4.0, "answer": 56}, {"id": "omni-01777", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 4.5, "answer":... | Discrete Mathematics / Combinatorics | 4.5 | 4.166667 | sum | subdomain | omni-math-composite | |
comp-train-00-00091 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Let \(P(x)\) be a quadratic polynomial with real coefficients. Suppose that \(P(1)=20, P(-1)=22\), and \(P(P(0))=400\). Compu... | 6523 | 3 | [{"id": "omni-02304", "subdomain": "Algebra / Algebra", "difficulty": 5.0, "answer": 2486}, {"id": "omni-04114", "subdomain": "Algebra / Algebra", "difficulty": 5.0, "answer": 4044}, {"id": "omni-01711", "subdomain": "Algebra / Algebra", "difficulty": 4.0, "answer": -7}] | Algebra / Algebra | 5 | 4.666667 | sum | subdomain | omni-math-composite | |
comp-train-00-00092 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Let $N$ be the number of distinct roots of \prod_{k=1}^{2012}\left(x^{k}-1\right)$. Give lower and upper bounds $L$ and $U$ o... | 1231467 | 2 | [{"id": "omni-03324", "subdomain": "Number Theory / Other", "difficulty": 5.0, "answer": 1231288}, {"id": "omni-01445", "subdomain": "Number Theory / Congruences", "difficulty": 4.0, "answer": 179}] | Number Theory / mixed | 5 | 4.5 | sum | domain | omni-math-composite | |
comp-train-00-00093 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Consider $9$ points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segmen... | 67983 | 2 | [{"id": "omni-03902", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 9.0, "answer": 33}, {"id": "omni-01287", "subdomain": "Discrete Mathematics / Combinatorics", "difficulty": 6.0, "answer": 67950}] | Discrete Mathematics / mixed | 9 | 7.5 | sum | domain | omni-math-composite | |
comp-train-00-00094 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Let $A B C$ be an acute scalene triangle with circumcenter $O$ and centroid $G$. Given that $A G O$ is a right triangle, $A O... | 294 | 2 | [{"id": "omni-01338", "subdomain": "Geometry / Plane Geometry", "difficulty": 5.25, "answer": 288}, {"id": "omni-00295", "subdomain": "Geometry / Plane Geometry", "difficulty": 7.0, "answer": 6}] | Geometry / Plane Geometry | 7 | 6.125 | sum | subdomain | omni-math-composite | |
comp-train-00-00095 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
In circle $\omega$, two perpendicular chords intersect at a point $P$. The two chords have midpoints $M_{1}$ and $M_{2}$ resp... | 685 | 3 | [{"id": "omni-02462", "subdomain": "Geometry / Plane Geometry", "difficulty": 4.0, "answer": 7}, {"id": "omni-02126", "subdomain": "Geometry / Plane Geometry", "difficulty": 4.0, "answer": 414}, {"id": "omni-00529", "subdomain": "Geometry / Plane Geometry", "difficulty": 5.0, "answer": 264}] | Geometry / Plane Geometry | 5 | 4.333333 | sum | subdomain | omni-math-composite | |
comp-train-00-00096 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Let $g_{1}(x)=\frac{1}{3}\left(1+x+x^{2}+\cdots\right)$ for all values of $x$ for which the right hand side converges. Let $g... | 8 | 2 | [{"id": "omni-02558", "subdomain": "Algebra / Intermediate Algebra", "difficulty": 5.0, "answer": 5}, {"id": "omni-01462", "subdomain": "Algebra / Algebra", "difficulty": 5.0, "answer": 3}] | Algebra / mixed | 5 | 5 | sum | domain | omni-math-composite | |
comp-train-00-00097 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Call an positive integer almost-square if it can be written as $a \cdot b$, where $a$ and $b$ are integers and $a \leq b \leq... | 130368 | 2 | [{"id": "omni-02500", "subdomain": "Number Theory / Factorization", "difficulty": 5.25, "answer": 130348}, {"id": "omni-02501", "subdomain": "Number Theory / Prime Numbers", "difficulty": 4.0, "answer": 20}] | Number Theory / mixed | 5.25 | 4.625 | sum | domain | omni-math-composite | |
comp-train-00-00098 | This task has 2 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 2 integer answers. Report only that sum.
Part 1:
Convex quadrilateral $B C D E$ lies in the plane. Lines $E B$ and $D C$ intersect at $A$, with $A B=2$, $A C=5, A D=200, A E=... | 545 | 2 | [{"id": "omni-02590", "subdomain": "Geometry / Plane Geometry", "difficulty": 5.0, "answer": 5}, {"id": "omni-02621", "subdomain": "Geometry / Plane Geometry", "difficulty": 5.0, "answer": 540}] | Geometry / Plane Geometry | 5 | 5 | sum | subdomain | omni-math-composite | |
comp-train-00-00099 | This task has 3 independent parts. Each part is a mathematics problem whose answer is an integer. The final answer to the task is the SUM of the 3 integer answers. Report only that sum.
Part 1:
Let $S=\left\{p_{1} p_{2} \cdots p_{n} \mid p_{1}, p_{2}, \ldots, p_{n}\right.$ are distinct primes and $\left.p_{1}, \ldots,... | 378532 | 3 | [{"id": "omni-00918", "subdomain": "Number Theory / Prime Numbers", "difficulty": 5.0, "answer": 512}, {"id": "omni-02064", "subdomain": "Number Theory / Prime Numbers", "difficulty": 5.0, "answer": 378000}, {"id": "omni-02501", "subdomain": "Number Theory / Prime Numbers", "difficulty": 4.0, "answer": 20}] | Number Theory / Prime Numbers | 5 | 4.666667 | sum | subdomain | omni-math-composite |
End of preview. Expand in Data Studio
math-team-composites
Composite competition-mathematics tasks for studying and training multi-agent coordination. Each task bundles 2 or 3 integer-answer problems from Omni-MATH (difficulty >= 4, one sub-domain where possible) into one task whose answer is the sum of the parts' answers. A team of agents can split the parts; a single agent must solve all of them within one budget.
Configs and splits
| config | train | validation | parts per task |
|---|---|---|---|
mixed_2_3_parts (default) |
10,000 | 500 | 2 or 3, drawn evenly (train 4,896 / 5,104; validation 224 / 276) |
three_parts |
10,000 | 500 | 3 |
Base problems: the 1,232 integer-answer Omni-MATH problems at difficulty >= 4 whose reference is a single closed-form expression (no yes/no questions, relations, sets, tuples, prose or non-unique answers), split into 1,045 train and 187 validation base problems. No base problem appears in both splits, in either config. The first 100 validation rows of each config are the evaluation samples used for the baselines below.
Fields
id,problem(the full composite statement: the parts, the instruction that each part's answer is an integer, and that the final answer is the sum),answer(the sum, as a string),k(number of parts)parts: JSON list of{id, subdomain, difficulty, answer}for the component Omni-MATH problemssubdomain(shared sub-domain, or<domain> / mixed),difficulty(max over parts),difficulty_mean,combine(sum),grouping(subdomainordomain),source,origin
Verification: compare the submitted expression with answer using Math-Verify
(integers, so exact match also works).
Baselines (100 validation tasks of each config, same tasks for every arm)
| arm | mixed 2/3 parts | three parts |
|---|---|---|
| gpt-oss-20b, one 4096-token call (medium effort) + 1024-token wrap-up | 19% | 4% |
| gpt-oss-20b, one 32k-token call (high effort) + wrap-up, pass@1 / pass@2 | 42% / 49% | 20% / 27% |
| team of 4 gpt-oss-20b, 3 board rounds, random submitter (one answer) | 44-48% | 19% |
| GPT-5.6 Sol, one 32k call (high) + wrap-up | see repo | 77% |
Environment, harness and analysis code: https://github.com/damanimehul/steganography (comms/tasks/math_team/).
Built 2026-10-27.
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