problem string | solution string | candidates list | tags list | metadata dict |
|---|---|---|---|---|
Let $ABC$ be an acute triangle with altitude $AD$ ( $D \in BC$ ). The line through $C$ parallel to $AB$ meets the perpendicular bisector of $AD$ at $G$ . Show that $AC = BC$ if and only if $\angle AGC = 90^{\circ}$ . | $\bullet$ $CA=CB:$ Let $E$ and $F$ be midpoints of $AD$ and $AB$ ,respectively. Since $GE||BC$ we get $F-E-G$ are collinear $\implies AF=FB=FD$ . $\angle GCA=\angle CAB=\angle CBA=\angle GFA \implies GCAF$ is cyclic $\implies \angle AGC=180-\angle CFA=180-90=90. \square$ $\bullet$ $\angle AGC=90:... | [
" $GM\\parallel BC, AB\\parallel BC$ , implies $AMCG$ is a parallelogram. $\\angle AGC=90^\\circ\\Leftrightarrow \\angle AMC=90^\\circ\\Leftrightarrow AC=BC$ since $M$ is midpoint of $AB$ .",
"Let $M$ be the midpoint of $\\overline{AB}$ and note $BCGM$ is a parallelogram. Then, $MD=MB=GC$ so $CDM... | [
"origin:aops",
"2022 Contests",
"2022 3rd Memorial "Aleksandar Blazhevski-Cane""
] | {
"answer_score": 130,
"boxed": false,
"end_of_proof": true,
"n_reply": 3,
"path": "Contest Collections/2022 Contests/2022 3rd Memorial "Aleksandar Blazhevski-Cane"/2759376.json"
} |
A $6 \times 6$ board is given such that each unit square is either red or green. It is known that there are no $4$ adjacent unit squares of the same color in a horizontal, vertical, or diagonal line. A $2 \times 2$ subsquare of the board is *chesslike* if it has one red and one green diagonal. Find the maximal po... | [] | [
"origin:aops",
"2022 Contests",
"2022 3rd Memorial "Aleksandar Blazhevski-Cane""
] | {
"answer_score": 0,
"boxed": false,
"end_of_proof": false,
"n_reply": 0,
"path": "Contest Collections/2022 Contests/2022 3rd Memorial "Aleksandar Blazhevski-Cane"/2759377.json"
} | |
Given an integer $n\geq2$ , let $x_1<x_2<\cdots<x_n$ and $y_1<y_2<\cdots<y_n$ be positive reals. Prove that for every value $C\in (-2,2)$ (by taking $y_{n+1}=y_1$ ) it holds that $\hspace{122px}\sum_{i=1}^{n}\sqrt{x_i^2+Cx_iy_i+y_i^2}<\sum_{i=1}^{n}\sqrt{x_i^2+Cx_iy_{i+1}+y_{i+1}^2}$ .
*Proposed by Mirko Petru... | We can use a similar argument as in the proof of rearrangement inequality. Letting $f(x,y)=\sqrt{x^2+Cxy+y^2}$ , it suffices to show the case $n=2$ , which corresponds to a single transposition in the general case.
Al we have to show that if $a<b$ and $c<d$ , then $$ \sqrt{a^2+Cac+c^2}+\sqrt{b^2+Cbd+d^2}<\sqrt{a... | [] | [
"origin:aops",
"2022 Contests",
"2022 3rd Memorial "Aleksandar Blazhevski-Cane""
] | {
"answer_score": 50,
"boxed": false,
"end_of_proof": false,
"n_reply": 1,
"path": "Contest Collections/2022 Contests/2022 3rd Memorial "Aleksandar Blazhevski-Cane"/2759383.json"
} |
Find all triplets of positive integers $(x, y, z)$ such that $x^2 + y^2 + x + y + z = xyz + 1$ .
*Proposed by Viktor Simjanoski* | <details><summary>Solution (using Vieta's Jumping Root Method)</summary>$\wedge$ means 'and'. $\mathbb{N*}$ means $\{n | n\in \mathbb{Z} \wedge n>0\}$ . $(*)$ stands for the equation $x^2+y^2+x+y+z=xyz+1$ .
Define $g(x,y):=\frac{x^2+y^2+x+y-1}{xy-1}$ .
WLOG assume $x\geq y$ . $\textbf{Case 1.}$ $y=1$ . $x^2+x+... | [
"<details><summary>Hint</summary>Vieta jumping. Solutions exists only for $z=7$ There are two series of solutions with first terms $1,2$ and $1,4$</details>",
"Vieta jumping method and pell equation.",
"what is your motivation to prove z=7 please?",
"Does anybody have a complete solution?\n"
] | [
"origin:aops",
"2022 Contests",
"2022 3rd Memorial "Aleksandar Blazhevski-Cane""
] | {
"answer_score": 128,
"boxed": false,
"end_of_proof": false,
"n_reply": 5,
"path": "Contest Collections/2022 Contests/2022 3rd Memorial "Aleksandar Blazhevski-Cane"/2759385.json"
} |
Find all positive integers $n$ such that the set $S=\{1,2,3, \dots 2n\}$ can be divided into $2$ disjoint subsets $S_1$ and $S_2$ , i.e. $S_1 \cap S_2 = \emptyset$ and $S_1 \cup S_2 = S$ , such that each one of them has $n$ elements, and the sum of the elements of $S_1$ is divisible by the sum of the el... | We claim the answer is all $n \not\equiv 5 \pmod 6$ . Let $\sum_{i \in S_1} i=A$ and $\sum_{i \in S_2} i=B$ . Then, $A+B=n(2n+1)$ and $A \mid B$ . Note that $A \geq 1+2+\ldots+n=\dfrac{n(n+1)}{2}$ and $B \leq 2n+(2n-1)+\ldots+(n+1)=\dfrac{n(3n+1)}{2}.$ Therefore, $B \leq \dfrac{n(3n+1)}{2} <\dfrac{3n(n+1)}{2} \... | [
"The answer is all $n \\not \\equiv 5\\pmod{6}$ .**Constraction for $n=2k$** : $S_1=\\{1,2,...,k\\}\\cup \\{3k+1,3k+2,...,4k\\}$ and $S_2=S\\setminus S_1$ .\nFor $n\\equiv 1,3 \\pmod{6}$ I will not give a construction but I will show that it's possible to construct $S_1$ and $S_2$ .\nLet $n=2k+1$ and le... | [
"origin:aops",
"2022 Contests",
"2022 3rd Memorial "Aleksandar Blazhevski-Cane""
] | {
"answer_score": 90,
"boxed": false,
"end_of_proof": false,
"n_reply": 2,
"path": "Contest Collections/2022 Contests/2022 3rd Memorial "Aleksandar Blazhevski-Cane"/2759387.json"
} |
Let $ABC$ be an acute triangle with incircle $\omega$ , incenter $I$ , and $A$ -excircle $\omega_{a}$ . Let $\omega$ and $\omega_{a}$ meet $BC$ at $X$ and $Y$ , respectively. Let $Z$ be the intersection point of $AY$ and $\omega$ which is closer to $A$ . The point $H$ is the foot of the altitude... | It's well known $XZ \perp BC$ . Let $AX$ and $HZ$ meet at $S$ , Note that $ZX || AH$ so $S$ lies on median of $AH$ in triangle $AYH$ so we must prove $IY$ is median of $AH$ . Note that $I$ is midpoint of $XZ$ and $AH || XZ$ so $IY$ is median of $AH$ .
we're Done. | [
"From \"Diameter of Incircle\" Lemma we know that $X-I-Z$ are collinear. So in $\\triangle AHY$ $YI$ is median and $ZX||AH$ . So from Ceva's Theorem we get $AX-HZ-IY$ are concurrent.",
" $XZ$ is diameter of $\\omega$ and $AH$ parallel $XZ$ .Remaning easy."
] | [
"origin:aops",
"2022 Contests",
"2022 3rd Memorial "Aleksandar Blazhevski-Cane""
] | {
"answer_score": 30,
"boxed": false,
"end_of_proof": false,
"n_reply": 3,
"path": "Contest Collections/2022 Contests/2022 3rd Memorial "Aleksandar Blazhevski-Cane"/2759390.json"
} |
We say that a positive integer $n$ is *memorable* if it has a binary representation with strictly more $1$ 's than $0$ 's (for example $25$ is memorable because $25=(11001)_{2}$ has more $1$ 's than $0$ 's). Are there infinitely many memorable perfect squares?
*Proposed by Nikola Velov* | $n^2=2^k \cdot a_k + ... + 2^1 \cdot a_1 + 2^0 a_0$ Next number $$ \boxed {(2^{k+2} + 1)n} $$ | [
"<blockquote>We say that a positive integer $n$ is *memorable* if it has a binary representation with strictly more $1$ 's than $0$ 's (for example $25$ is memorable because $25=(11001)_{2}$ has more $1$ 's than $0$ 's). Are there infinitely many memorable perfect squares?</blockquote>\nYes, there are .\n... | [
"origin:aops",
"2022 Contests",
"2022 3rd Memorial "Aleksandar Blazhevski-Cane""
] | {
"answer_score": 1006,
"boxed": true,
"end_of_proof": false,
"n_reply": 2,
"path": "Contest Collections/2022 Contests/2022 3rd Memorial "Aleksandar Blazhevski-Cane"/2759392.json"
} |
For any integer $n\geq1$ , we consider a set $P_{2n}$ of $2n$ points placed equidistantly on a circle. A *perfect matching* on this point set is comprised of $n$ (straight-line) segments whose endpoints constitute $P_{2n}$ . Let $\mathcal{M}_{n}$ denote the set of all non-crossing perfect matchings on $P_{2n... | [] | [
"origin:aops",
"2022 Contests",
"2022 3rd Memorial "Aleksandar Blazhevski-Cane""
] | {
"answer_score": 0,
"boxed": false,
"end_of_proof": false,
"n_reply": 0,
"path": "Contest Collections/2022 Contests/2022 3rd Memorial "Aleksandar Blazhevski-Cane"/2759397.json"
} | |
 These problems are copyright $\copyright$ [Mathematical Association of America](http://maa.org). | [] | [
"origin:aops",
"2022 Contests",
"2022 AIME Problems"
] | {
"answer_score": 0,
"boxed": false,
"end_of_proof": false,
"n_reply": 0,
"path": "Contest Collections/2022 Contests/2022 AIME Problems/1100560.json"
} | |
For any finite set $X$ , let $|X|$ denote the number of elements in $X.$ Define $$ S_n = \sum |A \cap B|, $$ where the sum is taken over all ordered pairs $(A, B)$ such that $A$ and $B$ are subsets of $\{1, 2, 3, …, n\}$ with $|A| = |B|.$ For example, $S_2 = 4$ because the sum is taken over the pai... | <blockquote>For any finite set $X$ , let $|X|$ denote the number of elements in $X.$ Define $$ S_n = \sum |A \cap B|, $$ where the sum is taken over all ordered pairs $(A, B)$ such that $A$ and $B$ are subsets of $\{1, 2, 3, …, n\}$ with $|A| = |B|.$ For example, $S_2 = 4$ because the sum is taken ... | [
"245, basically S_n = n(2n -2 choose n-1) from chairperson and vandermonde spam",
"proudest solve lesgo",
"Consider how many times any given number $k$ is counted in the intersection of $A, B$ , in the expression for $S_n$ . If $A, B$ each contain $r$ numbers, then it is counted ${n-1\\choose r-1}^2={... | [
"origin:aops",
"2022 Contests",
"2022 AIME Problems"
] | {
"answer_score": 1048,
"boxed": false,
"end_of_proof": false,
"n_reply": 48,
"path": "Contest Collections/2022 Contests/2022 AIME Problems/2777199.json"
} |
Let $S$ be the set of all rational numbers that can be expressed as a repeating decimal in the form $0.\overline{abcd},$ where at least one of the digits $a, b, c, $ or $d$ is nonzero. Let $N$ be the number of distinct numerators when numbers in $S$ are written as fractions in lowest terms. For example, bot... | The factors of $9999$ are $1, 3, 9, 11, 33, 99, 101, 303, 909, 1111, 3333, $ and $9999$ . For any integer in the range $[1, 9998]$ , it can be a numerator if there exists a factor of $9999$ that is relatively prime to that integer (because that factor can be its denominator). We now break this big interval into... | [
"395 gang anyone?",
"i got 449",
"answer is 392 from 6392 confirmed with code",
"<blockquote>answer is 392 from 6392 confirmed with code</blockquote>\n\nyeah same here I immediately wrote a Java code after the test \n\nsadge moment when you forget to delete the three multiples of $303$ in the $1111$ set :... | [
"origin:aops",
"2022 Contests",
"2022 AIME Problems"
] | {
"answer_score": 1188,
"boxed": true,
"end_of_proof": false,
"n_reply": 48,
"path": "Contest Collections/2022 Contests/2022 AIME Problems/2777200.json"
} |
Find the number of ordered pairs of integers $(a, b)$ such that the sequence $$ 3, 4, 5, a, b, 30, 40, 50 $$ is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression. | Clearly just picking from the set $\{3, 4, 5, 30, 40, 50\}$ we cannot find an arithmetic progression.
Case $1$ : The arithmetic progression contains only $a$ or only $b$ . Note that $6 \leq a \leq 28$ and $7 \leq b \leq 29$ .
Clearly $a = 6$ fails from $3, 4, 5, a$ . Next $a/b, 30, 40, 50$ causes $a ... | [
"Note that $7\\le a<b\\le 29$ and $a\\ne 20, b\\ne 20$ . The only other restrictions are $(7,9)$ , $(12,21)$ , and $(16,28)$ . So the answer is $\\binom{23}{2}-9-13-3=\\boxed{228}$ . ",
"I put 237 lmao\n\n(forgot to remove (20, 21), (20, 22), (20, 23), ..., (20, 29) ah stupid me)",
"I got 236 oof",
"Fo... | [
"origin:aops",
"2022 Contests",
"2022 AIME Problems"
] | {
"answer_score": 1074,
"boxed": false,
"end_of_proof": false,
"n_reply": 61,
"path": "Contest Collections/2022 Contests/2022 AIME Problems/2777203.json"
} |
Ellina has twelve blocks, two each of red $\left({\bf R}\right),$ blue $\left({\bf B}\right),$ yellow $\left({\bf Y}\right),$ green $\left({\bf G}\right),$ orange $\left({\bf O}\right),$ and purple $\left({\bf P}\right).$ Call an arrangement of blocks *even* if there is an even number of blocks between each... | Wow, this problem was actually so amazing and reminded me of why I enjoy comp math.**<span style="color:#f00">Claim:</span>** There exists a bijection between even arrangements with $n$ pairs of colored blocks and ways to order the evens and the odds (separately) from $1$ to $2n$ .
*Proof.* Label the ordering of t... | [
"Note that the even positions (2,4,6,8,10,12) have one of each color and same thing with odd positions. There are totally $\\frac{12!}{64}$ permutations. So the fraction is \\[\\frac{6!^2\\cdot 64}{12!}=\\frac{6!\\cdot 64}{7\\cdot 8\\cdot 9\\cdot 10\\cdot 11\\cdot 12}=\\frac{720\\cdot 8}{7\\cdot 9\\cdot 10\\cdot ... | [
"origin:aops",
"2022 Contests",
"2022 AIME Problems"
] | {
"answer_score": 1122,
"boxed": false,
"end_of_proof": false,
"n_reply": 53,
"path": "Contest Collections/2022 Contests/2022 AIME Problems/2777204.json"
} |
Let $a, b, c, d, e, f, g, h, i$ be distinct integers from $1$ to $9$ . The minimum possible positive value of $$ \frac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i} $$ can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | Note that $\frac{2\cdot 3\cdot 6-1\cdot 5\cdot 7}{4\cdot 8\cdot 9}=\frac{1}{288}$ . We claim this is the minimum, which gives an answer of $\boxed{289}$ .
Suppose there was something less. Then $abc-def=1$ .
If $9$ was in $a,b,c,d,e,f$ , then we would need $ghi=6\cdot 7\cdot 8$ . Now $a,b,c,d,e,f$ is some pe... | [
"Note that $(6,2,3,7,5,1,4,8,9)$ gives $\\tfrac{1}{288}$ , for an answer of $1+288=\\boxed{289}$ . Otherwise, if $abc - def = 2$ , then the minimum possible value is $\\tfrac{2}{7 \\cdot 8 \\cdot 9} = \\tfrac{1}{252}$ . ",
"i got 289? bsically let x = abc, y = def, then xy(x-y)/9! min which is x = 35 y = 36... | [
"origin:aops",
"2022 Contests",
"2022 AIME Problems"
] | {
"answer_score": 1052,
"boxed": false,
"end_of_proof": false,
"n_reply": 31,
"path": "Contest Collections/2022 Contests/2022 AIME Problems/2777205.json"
} |
Quadratic polynomials $P(x)$ and $Q(x)$ have leading coefficients of $2$ and $-2$ , respectively. The graphs of both polynomials pass through the two points $(16,54)$ and $(20,53)$ . Find ${P(0) + Q(0)}$ . | <blockquote>We have $P(x)=2x^2+ax+b$ . So $512+16a+b=54\implies 16a+b=-458$ . Also, $800+20a+b=53$ , so $20a+b=-747$ . Thus, $4a=-289$ . So $-1156+b=-458\implies b=698$ .
Also, $Q(x)=-2x^2+cx+d$ . So $-512+16c+d=54\implies 16c+d=566$ . Also, $-800+20c+d=53\implies 20c+d=853$ . So $4c=287$ . So $1148+d=566\i... | [
"Mine.\n\n<details><summary>Solution</summary>The polynomial $R(x) := P(x) + Q(x)$ is linear, and we may compute $R(16) = 108$ and $R(20) = 106$ . Hence $R(0) = \\boxed{116}$ .</details>",
"<details><summary>other sol</summary>The line that passes through both points is $y=-\\frac{1}{4}x+58$ .\n Hence, ... | [
"origin:aops",
"2022 Contests",
"2022 AIME Problems"
] | {
"answer_score": 1032,
"boxed": true,
"end_of_proof": false,
"n_reply": 37,
"path": "Contest Collections/2022 Contests/2022 AIME Problems/2777211.json"
} |
Find the three-digit positive integer $\underline{a} \ \underline{b} \ \underline{c}$ whose representation in base nine is $\underline{b} \ \underline{c} \ \underline{a}_{\hspace{.02in}\text{nine}}$ , where $a$ , $b$ , and $c$ are (not necessarily distinct) digits. | We claim $\boxed{227}$ works.
Proof: $227=2\cdot 81+7\cdot 9+2$ . $\blacksquare$ . | [
" $227$ works I think. What I did was set up the equation $99a=71b+8c$ and casework on $a$ .",
"<blockquote> $227$ works I think.</blockquote>\n\nGot that too",
"<blockquote> $227$ works I think. What I did was set up the equation $99a=71b+8c$ and casework on $a$ .</blockquote>\n\ntaking mod 9 works be... | [
"origin:aops",
"2022 Contests",
"2022 AIME Problems"
] | {
"answer_score": 1106,
"boxed": false,
"end_of_proof": false,
"n_reply": 41,
"path": "Contest Collections/2022 Contests/2022 AIME Problems/2777212.json"
} |
Let $x$ , $y$ , and $z$ be positive real numbers satisfying the system of equations
\begin{align*}
\sqrt{2x - xy} + \sqrt{2y - xy} & = 1
\sqrt{2y - yz} + \hspace{0.1em} \sqrt{2z - yz} & = \sqrt{2}
\sqrt{2z - zx\vphantom{y}} + \sqrt{2x - zx\vphantom{y}} & = \sqrt{3}.
\end{align*}Then $\big[ (1-x)(1-y)(1-z) \big] ^2... | Favorite problem on the test. Extremely clean. (Solution close to that in post #2)
First, we note that we can let a triangle exist with side lengths $\sqrt{2x}$ , $\sqrt{2z}$ , and opposite altitude $\sqrt{xz}$ . This shows that the third side, which is the nasty square-rooted sum, is going to have the length equal... | [
"Magical solution communicated to me by a girl in my school who doesn't even do competition math and got this during the test. Let $x=2\\sin^2\\alpha, y=2\\sin^2\\beta, z=2\\sin^2\\theta$ . The given conditions rewrite themselves as:\n\\begin{align*}\n2\\sin(\\alpha+\\beta)&=1 \n2\\sin(\\beta+\\theta)&=\\sqrt{2} \... | [
"origin:aops",
"2022 Contests",
"2022 AIME Problems"
] | {
"answer_score": 1158,
"boxed": true,
"end_of_proof": false,
"n_reply": 43,
"path": "Contest Collections/2022 Contests/2022 AIME Problems/2777215.json"
} |
In isosceles trapezoid $ABCD$ , parallel bases $\overline{AB}$ and $\overline{CD}$ have lengths $500$ and $650$ , respectively, and $AD=BC=333$ . The angle bisectors of $\angle{A}$ and $\angle{D}$ meet at $P$ , and the angle bisectors of $\angle{B}$ and $\angle{C}$ meet at $Q$ . Find $PQ$ . | <blockquote><blockquote>Diagram:
[asy]
unitsize(0.016cm);
pair A = (-300,324.4);
pair B = (300, 324.4);
pair C = (375, 0);
pair D = (-375, 0);
draw(A--B--C--D--cycle);
pair W = (-42,0);
pair X = (42, 0);
pair Y = (-33,324.4);
pair Z = (33,324.4);
pair P = (-171, 162.2);
pair Q = (171, 162.2);
dot(P);
dot(Q);
draw(A--... | [
"Solution (related to the title):\n\nTranslate points $B, Q,$ and $C$ by $PQ$ units to the left, as shown. Let $PQ = x$ .\n[asy]\nsize(250);\nlabel((0,0), \"D\", SW);\nlabel((1.166666666, 6), \"A\", NW);\nlabel((4,2.6), \"P, Q'\", S);\nlabel((8,3), \"Q\", S);\nlabel((12,0), \"C\", SE);\nlabel((10.833333, 6),... | [
"origin:aops",
"2022 Contests",
"2022 AIME Problems"
] | {
"answer_score": 1178,
"boxed": true,
"end_of_proof": false,
"n_reply": 74,
"path": "Contest Collections/2022 Contests/2022 AIME Problems/2777216.json"
} |
Given $\triangle ABC$ and a point $P$ on one of its sides, call line $\ell$ the splitting line of $\triangle ABC$ through $P$ if $\ell$ passes through $P$ and divides $\triangle ABC$ into two polygons of equal perimeter. Let $\triangle ABC$ be a triangle where $BC = 219$ and $AB$ and $AC$ are po... | <blockquote>Given $\triangle ABC$ and a point $P$ on one of its sides, call line $\ell$ the splitting line of $\triangle ABC$ through $P$ if $\ell$ passes through $P$ and divides $\triangle ABC$ into two polygons of equal perimeter. Let $\triangle ABC$ be a triangle where $BC = 219$ and $AB$ and ... | [
"Consider the splitting line through $M$ . Extend $D$ on ray $BC$ such that $CD=CA$ . Then the splitting line bisects segment $BD$ , so in particular it is the midline of triangle $ABD$ and thus it is parallel to $AD$ . But since triangle $ACD$ is isosceles, we can easily see $AD$ is parallel to the a... | [
"origin:aops",
"2022 Contests",
"2022 AIME Problems"
] | {
"answer_score": 1198,
"boxed": true,
"end_of_proof": false,
"n_reply": 39,
"path": "Contest Collections/2022 Contests/2022 AIME Problems/2777218.json"
} |
End of preview. Expand in Data Studio
AoPS: Art of Problem Solving Competition Mathematics
Dataset Description
This dataset is a collection of 80,661 competition mathematics problems and solutions obtained from the Art of Problem Solving (AoPS) community wiki and forums. It covers a wide range of mathematical contests and olympiads, including problems from events such as AIME, BAMO, IMO, and various national and memorial competitions.
The dataset was curated by AI-MO (Project Numina), an initiative focused on building AI systems capable of mathematical reasoning at the olympiad level.
Dataset Structure
Fields
| Column | Type | Description |
|---|---|---|
problem |
string |
The mathematical problem statement, typically formatted in LaTeX. |
solution |
string |
A solution or proof for the problem. May be empty for some entries. |
candidates |
list[string] |
Alternative or candidate solutions contributed by the community. |
tags |
list[string] |
Metadata tags indicating the origin, contest name, and year (e.g., "origin:aops", "2022 AIME Problems"). |
metadata |
dict |
Additional metadata about the problem (see below). |
Metadata Fields
| Field | Type | Description |
|---|---|---|
answer_score |
int64 |
Community score or rating of the answer. |
boxed |
bool |
Whether the answer contains a boxed final result (e.g., \boxed{42}). |
end_of_proof |
bool |
Whether the solution includes a complete proof ending. |
n_reply |
int64 |
Number of community replies or comments on the problem thread. |
path |
string |
Source path in the AoPS collection (e.g., Contest Collections/2022 Contests/...). |
Splits
| Split | Examples |
|---|---|
train |
80,661 |
Example
{
"problem": "Let $ABC$ be an acute triangle with altitude $AD$ ($D \\in BC$). The line through $C$ parallel to $AB$ meets the perpendicular bisector of $AD$ at $G$. Show that $AC = BC$ if and only if $\\angle AGC = 90°$.",
"solution": "...",
"candidates": ["..."],
"tags": ["origin:aops", "2022 Contests", "2022 3rd Memorial \"Aleksandar Blazhevski-Cane\""],
"metadata": {
"answer_score": 130,
"boxed": false,
"end_of_proof": true,
"n_reply": 3,
"path": "Contest Collections/2022 Contests/2022 3rd Memorial .../2759376.json"
}
}
Topic Coverage
Problems span a broad range of competition mathematics topics, including:
- Geometry -- triangle properties, cyclic quadrilaterals, angle chasing
- Number Theory -- divisibility, modular arithmetic, Diophantine equations
- Algebra -- inequalities, polynomials, functional equations
- Combinatorics -- counting, graph theory, board coloring problems
Usage
from datasets import load_dataset
dataset = load_dataset("AI-MO/aops")
# Access a problem
print(dataset["train"][0]["problem"])
print(dataset["train"][0]["solution"])
Intended Use
- Training and evaluating mathematical reasoning models
- Benchmarking LLMs on competition-level mathematics
- Studying solution quality and problem difficulty distributions
- Building retrieval-augmented generation (RAG) systems for math tutoring
Source
All problems and solutions originate from the Art of Problem Solving community.
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