agent_ref dict | problem stringlengths 33 2.28k | responses_create_params dict | license stringclasses 1
value | uuid stringlengths 36 36 | metadata listlengths 0 0 | source stringclasses 1
value | dataset stringclasses 1
value |
|---|---|---|---|---|---|---|---|
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute triangle with circumcircle $(O)$. Let $BF$ and $CE$ be the internal angle bisectors of $\angle B$ and $\angle C$, meeting sides $AC$ and $AB$ at $F$ and $E$, respectively, and intersecting at $I$ (the incenter). Extend line $EF$ to intersect $(O)$ again at points $M$ and $N$. Let $P\neq I$ be the ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 27c35773-d786-5589-ae1a-3cc9e4b79aa9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine whether there exists a polynomial $P(x,y,z)\in\mathbb Z[x,y,z]$ such that for every positive integer $n$, $n$ is not a perfect square if and only if there exist positive integers $x,y,z$ with $P(x,y,z)=n$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 914617aa-197b-5636-a98a-409b9da7e0c8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b\ge 0$ be real numbers such that $a^6+b^6=2$. Prove that
\[a+b\le a^5+b^5.\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 3ccc4db2-b7cf-56ee-bc43-64dd87e99858 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | There are several scientists collaborating in Niichavo. During an 8‑hour working day each scientist may visit the cafeteria any number of times. For any two distinct scientists, the total amount of time during which exactly one of the two is in the cafeteria (i.e. the measure of the symmetric difference of their cafete... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 37036a55-0141-5c0e-8ccc-9b36d69fb0d8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Vasya has a calculator that works with ordered pairs of integers. From a pair \((x,y)\) the calculator can produce either the pair \((x+y,\;x)\) or the pair \((2x+y+1,\;x+y+1)\). Initially the pair \((1,1)\) is on the calculator. Prove that for every natural number \(n\) there exists exactly one reachable pair of the f... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 36ed3b9c-fe2f-5d8e-8901-d4352b9cda17 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying each of the following equations:
1) $f(xf(x) + f(y)) = f(x)^2 + y$ for all real numbers $x,y$.
2) $(f(x) + f(y))(f(u) + f(v)) = f(xu - yv) + f(xv + yu)$ for all real numbers $x,y,u,v$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ac4d45cc-d019-573d-bfef-9750d67b337a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions \(f:\mathbb{Q}[x]\to\mathbb{R}\) such that
(a) for all \(P,Q\in\mathbb{Q}[x]\), \(f(P\circ Q)=f(Q\circ P)\) (where \(P\circ Q\) denotes \(P(Q(x))\));
(b) for all \(P,Q\in\mathbb{Q}[x]\) with \(PQ\neq0\), \(f(P\cdot Q)=f(P)+f(Q)\). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | bf68ee57-e99a-522c-8b96-c37c63207c24 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In free space, two identical particles A and B (each of mass m) are rigidly attached at the ends of a mass‑less rod of length $r_0$, forming a straight line. A third identical particle C (also of mass m) is placed at a distance $r_0$ from each of A and B so that the three particles form an equilateral triangle of side ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 745e8e2e-3bf3-5483-88ab-c66a17389981 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that for any positive integer $n$, the product $n(n+1)(n+2)(n+3)$ is never a perfect cube. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ca1ffb29-1a0c-55d6-9b03-1df6a53c1162 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n\ge 6$ be an integer. Determine the smallest integer $f(n)$ such that every simple undirected graph with $n$ vertices and at least $f(n)$ edges necessarily contains two distinct cycles that have no common vertex (i.e., two vertex‑disjoint cycles). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 9f0a9473-19d2-5a84-b92c-fdd392ff8f90 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For all real numbers $x,\;y,\;z$, prove that
\[
3\bigl(x^4+y^4+z^4-x^3y-y^3z-z^3x\bigr)\;\ge\;x^2(y-z)^2+y^2(z-x)^2+z^2(x-y)^2\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | d1a6a927-6991-513d-b4f3-6c162d443221 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n$ be a fixed natural number. Find all $n$‑tuples of natural numbers $(a_1,a_2,\dots,a_n)$ that are pairwise distinct and pairwise coprime and satisfy
\[
a_1+a_2+\cdots+a_n\;\Big|\;a_1^i+a_2^i+\cdots+a_n^i\qquad\text{for every }i=1,2,\dots,n\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | d4b91e6f-047c-5006-aea5-d17f4a69b770 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(a,b,c>0\). Prove that
\[
\sqrt[3]{\frac{a}{\,b+c\,}}+\sqrt[3]{\frac{b}{\,c+a\,}}+\sqrt[3]{\frac{c}{\,a+b\,}}>2\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 9e8372a6-f047-5046-ae4b-30f84606a576 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In a group of $k$ people, some are acquainted with each other and some are not. Each evening one person invites all of his acquaintances to a party and introduces any two of them who are not yet acquainted (i.e., after the party all of his acquaintances become pairwise acquainted). Suppose that after each person has ar... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 0bfc1884-4de4-592b-bb3f-b71f32ae6f7b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ satisfy
\[
a^2b^2+b^2c^2+c^2a^2=3.
\]
Prove that
\[
(a^2-a+1)(b^2-b+1)(c^2-c+1)\ge 1.
\]
Equality occurs when $a=b=c=1$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 5b20f834-6fd6-54e8-8325-3163decf5539 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\ge 0$ satisfy $a^{2}+b^{2}+c^{2}=3$. Prove that
\[
\frac{a+b+c}{3}\;\ge\;\Bigg(\frac{4+2abc}{\,7-abc\,}\Bigg)^{k},
\qquad\text{where}\qquad
k=\frac{\ln\sqrt{3}-\ln3}{\ln4-\ln7}\approx0.981577281.
\]Equality occurs at $a=b=c=1$ and this $k$ is the smallest exponent for which the inequality holds. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | f89b0ce7-50b0-5792-8a5d-a195b242bfb6 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | A lattice polytope $P$ in the plane is a polygon whose vertices are integer lattice points $(n,m)$ with $n,m\in\mathbb Z$. Let $E$ denote the number of lattice points lying on the edges of $P$ (including the vertices) and $I$ denote the number of lattice points lying strictly inside $P$. It is known that there exist ra... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 7ec05ab9-5819-54ee-a194-6952172ea34b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an isosceles triangle with $AB=AC$, inscribed in a circle $\omega$. Let the $B$‑symmedian of $\triangle ABC$ intersect $\omega$ again at $D$. Let $\Gamma_{1}$ be the circle through $C$ and $D$ that is tangent to line $BC$ at $C$, and let $\Gamma_{2}$ be the circle through $A$ and $D$ that is tangent to lin... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 0b6d5c07-8988-520b-87ce-5db219e22c91 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a trapezoid with $AB\parallel CD$ and $AB<CD$. Let $P$ be the intersection of the lines $AD$ and $BC$. Suppose a point $Q$ lies inside $ABCD$ and satisfies
\[
\angle QAB=\angle QDC=90^\circ-\angle BQC.
\]
Prove that
\[
\angle PQA=2\angle QCD.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 0b17ff37-719e-5f5d-875f-97128a99ee1b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\ge 0$ satisfy $a+b+c=\frac{7}{3}$. Prove that
\[
\bigl(ab+bc+ca+abc\bigr)\left(\frac1{(a+b)^3}+\frac1{(b+c)^3}+\frac1{(c+a)^3}\right)\ge\frac{51}{28}.
\]
Equality holds for $a=b=c=\frac79$ or for permutations of $(0,\frac76,\frac76)$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | e1ab5de2-bc34-59f6-8945-5976d887bb3e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a fixed triangle. Let $D$ be an arbitrary point inside $\triangle ABC$ such that the angle $\angle BAD$ is fixed. Draw the line through $D$ parallel to $BC$; it meets the circumcircle of $ABC$ again at $E$ and $F$. Prove that the tangent to the circumcircle of $ADE$ at $E$ and the tangent to the circumcirc... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 83da256d-2ca1-5254-acda-c21f49c18d5a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle. Denote by $I$ its incenter and by $(I)$ its incircle, which touches $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Let $O$ be the circumcenter of $ABC$. Let $P$ be a point such that
\[
\angle IDO = \angle IAP\quad\text{and}\quad \angle PCI = \angle OFI.
\]
Prove that
\[
\angle IBP = \angle... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 57655578-2e63-5341-9005-8b0dcc636631 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be positive numbers such that $a^4+b^4+c^4=a+b+c$. Prove that
\[
\frac{a}{c^2(b^2+ca)}+\frac{b}{a^2(c^2+ab)}+\frac{c}{b^2(a^2+bc)}\ge\frac32.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 784e8c28-c229-532a-9702-d3a7394f3c73 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\ge 0$ be real numbers such that $a+b+c=2$. Prove that
\[
a^{2}+b^{2}+c^{2}\;\ge\;2\bigl(a^{3}b^{3}+b^{3}c^{3}+c^{3}a^{3}\bigr)+9a^{2}b^{2}c^{2}.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | beb80d97-9a2a-5b42-9a25-19b402d5652c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all pairs of positive integers \((m,n)\) such that an \(m\times n\) rectangular board can be tiled with L‑shaped trominoes (three unit squares forming an “L”, allowed to be rotated). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ecde9f63-8816-5f1e-8f23-9b35837eb212 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $G$ be a group with $n$ elements. Let $A=\{x\in G\mid x^{2}=e\}$ and $k=|A|$. Define $f(n)$ to be the smallest integer $k$ such that every group $G$ of order $n$ satisfying $|A|\ge k$ must be abelian. Find $f(n)$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ccd6792a-076e-545a-a2a2-587d69f7e084 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | The sequence $\{a_n\}_{n\ge1}$ is defined by $a_1=2,\;a_2=12$ and
$$a_{n+1}=6a_n-a_{n-1}\qquad\text{for every integer }n\ge2.$$
Prove that no term $a_n$ is equal to a perfect power greater than one; i.e. there do not exist integers $m>1$ and $k\ge2$ such that $a_n=m^k$ for any $n\ge1\,. $ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 8c3d208f-3967-5b94-9375-dc20d0372f08 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\ge0$ with $a+b+c=3$. Prove that
\[
\frac{3}{1+2^{-\frac{2}{3}}}\;\le\;
\frac{a}{\,1+2^{-\frac{2}{3}}ab\,}
+\frac{b}{\,1+2^{-\frac{2}{3}}bc\,}
+\frac{c}{\,1+2^{-\frac{2}{3}}ca\,}
\;\le\;3.
\]
Equality in the left‑hand inequality occurs for $a:b:c=1:1:1$ and for $a:b:c=0:1:(\sqrt[3]{4}-\sqrt[3]{2})$ (and c... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 9c4c3882-ddf4-538f-83ab-5e05660ce4d0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | The points $A,B,K,L,X$ lie on a circle $\Gamma$ in that order and the arcs $\widehat{BK}$ and $\widehat{KL}$ are equal. Let $\omega_{1}$ be the unique circle passing through $A$ and tangent to line $BK$ at $B$; $\omega_{1}$ meets the segment $KX$ at the two points $P$ and $Q$. Let $\omega_{2}$ be the unique circle pass... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 094f397e-7fe1-5b75-8409-411b525ef955 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | If \(a,\,b,\,c\) are non‑negative real numbers such that \(ab+bc+ca>0\), prove that
\[
\frac{ab}{(a+b)^2}+\frac{bc}{(b+c)^2}+\frac{ca}{(c+a)^2}+\frac54\;\ge\;\frac{6\,(ab+bc+ca)}{(a+b+c)^2}.
\]
Equality occurs when \(a=b=c\) or when one of the variables is zero and the other two are equal. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | cb61358f-7ab0-56ef-bf3d-04573f4c0825 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | The king assembled 300 wizards. There are 25 known hat colors. Each wizard receives a hat of one of these colors, and the 25 numbers of hats of each color are all distinct (the wizards know this fact). Each wizard can see the hats of all other wizards but not his own, and all wizards simultaneously guess the color of t... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 7fee97e7-36bf-5ebf-95cc-36e76312db90 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given positive real numbers $a,b,c$ that are the side lengths of a triangle (i.e. $a,b,c>0$ and $a+b>c$, $b+c>a$, $c+a>b$), prove that
\[
a^{2}+b^{2}+c^{2}+a+b+c\;\ge\;3\sqrt[3]{\,abc\bigl(a^{2}+b^{2}+c^{2}+abc+1\bigr)+a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2}\,}.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 2c64f34e-43da-5c83-bed1-d4b53ae78ebf | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that
\[f\big(x^{2}+f(y)\big)=\big(f(x)+y^{2}\big)^{2}\]
for all real numbers $x$ and $y\,. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 652c778d-44e1-595e-80d9-83dce021c71a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Fix an odd integer $n>1$. For a permutation $p$ of the set $\{1,2,\dots,n\}$, let $S$ be the number of pairs of indices $(i,j)$ with $1\le i\le j\le n$ such that the sum $p_i+p_{i+1}+\dots+p_j$ is divisible by $n$. Determine the maximum possible value of $S$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 17c496fe-d4ca-5f22-a221-cce619346c45 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In triangle $ABC$, three congruent circles of radius $r$ are placed in the three corners: each circle is tangent to the two sides meeting at a vertex of $ABC$. Let $MNK$ and $PQR$ be two triangles such that
* each vertex of $MNK$ lies on a different side of $ABC$, and each side of $MNK$ is tangent to a different one... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | fe85f505-8b6a-5247-9ce4-066f724441e9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Each integer is colored with one of three colors. Prove that there exist two distinct integers of the same color whose difference is a perfect square. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 71bf3f9f-81d8-5c70-a64e-ca7a5545c5ab | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For every natural number $n$, prove that there exists a circle in the plane whose circumference contains exactly $n$ points with integer coordinates (lattice points). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 93f98bb6-8d2f-55e5-8cfa-e5fc581f5245 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be nonnegative real numbers such that $a^{2}+b^{2}+c^{2}=3$. Prove that
(a) $\,abc+\dfrac{5}{3}\min\{(a-b)^{2},(b-c)^{2},(c-a)^{2}\}\le 1;$
(b) $\,abc+|(a-b)(b-c)(c-a)|\le 1.$
When does equality occur? | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 2b79cae8-69c5-5bdb-a109-ae54ad63455f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $C_1, C_2, C_3, C_4$ be circles such that each consecutive pair of circles are tangent: $C_1$ and $C_2$ are tangent at $A$, $C_2$ and $C_3$ at $B$, $C_3$ and $C_4$ at $C$, and $C_4$ and $C_1$ at $D$. Prove that the quadrilateral $ABCD$ is cyclic. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a01dcecc-cbc3-51c8-8465-5232700d8de3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that a field K (commutative or not) cannot be expressed as the union of three proper subfields; i.e., there do not exist subfields K₁, K₂, K₃ ⊂ K with Kᵢ ≠ K such that K = K₁ ∪ K₂ ∪ K₃. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | bf047a55-a1de-5768-afbc-bc4dd53d0813 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle and let $P$ be a point on the segment $BC$. Let $X$ be the midpoint of the arc $AB$ of the circumcircle of $\triangle APB$ that lies inside $\triangle ABC$, and let $Y$ be the midpoint of the arc $AC$ of the circumcircle of $\triangle APC$ that lies inside $\triangle ABC$. Prove that the circumc... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 88ed0bf2-7829-5fc3-ba72-440af8671d7c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\triangle ABC$ be an isosceles triangle with $AC=BC$, and let $\Omega$ be its circumcircle. Let $M$ be a point on side $BC$. Let $N$ be a point on ray $AM$ beyond $M$ (so that $M$ lies between $A$ and $N$) such that $AN=AC$. Let $(CMN)$ be the circumcircle of $\triangle CMN$, and let $P\neq C$ be its second inters... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | fd0833bb-14d5-52a7-91e8-3bccb854a0c4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | If $a,b,c>0$ and $a+b+c=3$, prove that
\[
\frac{a}{3a+b^{2}}+\frac{b}{3b+c^{2}}+\frac{c}{3c+a^{2}}\le\frac34.
\]
Equality should be determined. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 6bbf22fd-bdd8-5f32-95e1-ac3a5e951ed6 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $E$ be a point on the internal $A$‑bisector of $\triangle ABC$ lying in the half‑plane determined by line $BC$ that does **not** contain $A$. Let $(E)$ be the circle centered at $E$ and tangent to $BC$. The two external common tangents to the circumcircle $(ABC)$ and the circle $(E)$ meet $BC$ at points $P$ and $... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ca8b4130-82db-530a-b5cf-06fbce4469b3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute‑angled triangle with $AC<BC$. A circle passing through $A$ and $B$ meets the segments $AC$ and $BC$ again at $A_1$ and $B_1$, respectively. The circumcircles of $A_1B_1C$ and $ABC$ intersect again at $P\neq C$. The lines $AB_1$ and $A_1B$ intersect at $S$. Let $Q$ and $R$ be the reflections of $S$... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | edd8577e-a32c-5907-aa64-d70c5ad01c6e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be positive real numbers. Prove that
(a) \[
\frac{1}{a^{2}+ab+b^{2}}+\frac{1}{b^{2}+bc+c^{2}}+\frac{1}{c^{2}+ca+a^{2}}\ge \frac{9}{(a+b+c)^{2}}.
\]
(b) \[
\frac{1}{a^{2}+pab+b^{2}}+\frac{1}{b^{2}+pbc+c^{2}}+\frac{1}{c^{2}+pca+a^{2}}\ge \frac{27}{(2+p)(a+b+c)^{2}}
\qquad\text{for }p\ge\frac78 .
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 402d826c-6cf2-575f-8dc3-372f9cb28f22 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given a triangle \(T\), let \(l\) be the maximum possible side length of a square whose four vertices lie on the sides of \(T\) (an inscribed square), and let \(L\) be the minimum possible side length of a square that contains \(T\) with each vertex of \(T\) on a side of the square (a circumscribed square). Prove that ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 225af975-e2c7-57fe-86cc-89ccbc0ffac3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $A\in M_n(\mathbb{Z})$ be such that the sequence $(A^p)_{p\in\mathbb N_{\ge1}}$ is bounded (with respect to any matrix norm).
i) Prove that there exists a positive integer $k$ such that $A^k$ is a projection, i.e. $(A^k)^2=A^k$.
ii) For $A\in M_{20}(\mathbb{Z})$ define
\[
k(A)=\min\{\,k\in\mathbb N_{\ge1}\m... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 7d90fe7b-53d8-586b-acbb-a6aff21c0493 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | A natural number $m$ is called **ziba** if every integer $n$ with $1\le n\le m$ can be expressed as a sum of distinct positive divisors of $m$. Prove that there exist infinitely many natural numbers of the form $k^2+k+2022$ that are ziba. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 4342f142-4b30-5da3-814a-929420407568 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\omega$ be a circle and $\omega_{1}$ a circle internally tangent to $\omega$ at $T$. Let $AB$ be a chord of $\omega$ such that $\omega_{1}$ meets $AB$ at $M$. Let $N$ be the second intersection of the line $TM$ with $\omega$. From $N$ draw the two tangents to $\omega_{1}$, touching $\omega_{1}$ at $C$ and $D$. Def... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | e9880899-275c-5901-bebc-4501b888811b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,\;b,\;c\ge0$. Prove the following three inequalities (cyclic sum means the sum over $(a,b),(b,c),(c,a)$):
\[
\sum_{\text{cyc}}\sqrt{a^{2}+ab+b^{2}}\;\le\;\sqrt{\sum_{\text{cyc}}\bigl(5a^{2}+4ab\bigr)},
\tag{1}
\]
\[
\sum_{\text{cyc}}\sqrt{4a^{2}+ab+4b^{2}}\;\le\;\sqrt{\sum_{\text{cyc}}\bigl(22a^{2}+5ab\bigr)},
\... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 42d1e717-d024-508f-ae4b-da7e1b6c8e9d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Denote $\mathbb R^*=\mathbb R\setminus\{0\}$. Find all functions $f:\mathbb R^*\to\mathbb R^*$ such that
$$f(xy)\,\bigl(f(x)+f(y)\bigr)=f(x+y)\,f(x)\,f(y)$$
for all $x,y\in\mathbb R^*$ with $x+y\neq0\,. $ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 41a6ca23-a33f-56f6-b274-d1859f6b6022 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a scalene triangle. Let $l_0$ be the tangent to the circumcircle of $ABC$ at $A$. Let $l$ be a variable line parallel to $l_0$, intersecting $AB$ at $X$ and $AC$ at $Y$. Let $T=BY\cap CX$ and let $Z$ be the second intersection of $AT$ with the circumcircle of $ABC$. Prove that as $l$ varies, the circumcirc... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 87cec194-98e7-5685-b963-120fc2650b57 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\omega_1$ and $\omega_2$ be two circles intersecting at points $A$ and $A'$, and let $T_1T_2$ be their external common tangent, touching $\omega_1$ at $T_1$ and $\omega_2$ at $T_2$. Choose any point $B_1$ on $\omega_1$ and any point $B_2$ on $\omega_2$ such that $A,B_1,B_2$ are not collinear. Let $\omega$ be the c... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | cd062d1a-2816-549a-bfaa-908cf9d99a50 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:(0,1)\to(0,1)$ such that $f\!\left(\tfrac12\right)=\tfrac12$ and
\[
\bigl(f(ab)\bigr)^{2}=\bigl(a\,f(b)+f(a)\bigr)\bigl(b\,f(a)+f(b)\bigr)
\qquad\text{for all }a,b\in(0,1).
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | d444e8b5-784f-5f6a-9a26-84ed525843f1 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,y,z\ge 0$. Prove that
\[
\sqrt{\frac{7x}{\,x+3y+3z\,}}+\sqrt{\frac{7y}{\,y+3z+3x\,}}+\sqrt{\frac{7z}{\,z+3x+3y\,}}\le 3.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | e0c44366-6d74-51f3-af33-ad483cc7da23 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(a,b,c\) be non‑negative real numbers. Prove that
\[
\sum_{\text{cyc}} (23a-5b-c)\,(a-b)^{2}\,(a+b-3c)^{2}\ge 0.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 6cd33590-d392-543c-84bc-5421633895ef | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given positive real numbers $a,b,c$ satisfying $ab+bc+ca=1$, prove that
\[
\frac{(1+a^{2})^{2}}{b^{2}+c^{2}}+\frac{(1+b^{2})^{2}}{c^{2}+a^{2}}+\frac{(1+c^{2})^{2}}{a^{2}+b^{2}}\ge 8.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 8bf295af-8045-5a0f-a1ee-3c729cfe87e8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | There are \(n\ge 4\) people. Between any two people at most one handshake can occur. For any four distinct people \(A,B,C,D\) such that \(A\) shakes hands with \(B\), \(B\) shakes hands with \(C\), and \(C\) shakes hands with \(D\), at least one of the handshakes \(A\!-\!C\), \(A\!-\!D\), \(B\!-\!D\) also occurs. Prove... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | c0a60974-7d00-5121-be82-08f1d4f9ec1b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,y,z\ge0$. Prove each of the following inequalities:
1. \[(x^3+y^3+z^3)(xy^2+yz^2+zx^2)-(xy+yz+zx)(xz^3+x^3y+y^3z)\ge0;\]
2. \[(xy^2+yz^2+zx^2)(x^2y+y^2z+z^2x)-(x+y+z)(x^2+y^2+z^2)xyz\ge0;\]
3. \[\frac{9}{8}(y+z)(z+x)(x+y)(x^3+y^3+z^3)-(x+y+z)(x^2+y^2+z^2)(xy^2+yz^2+zx^2)\ge0.\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 01990889-ec94-50c4-839d-c4ccc7e0f9da | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Define \(f:\mathbb{N}\to\mathbb{N}\) by
\[
f(n)=\sum_{\substack{t_1,\dots,t_n\ge0\\\sum_{j=1}^{n}j\,t_j=n}}
\frac{\bigl(1+\sum_{i=1}^{n}t_i\bigr)!}{\bigl(1+t_1\bigr)\,\prod_{i=1}^{n}t_i!},
\qquad (f(0)=1).
\]
Let \(p\) be a prime with \(p>3\). Prove that
\[
\sum_{1\le i<j<k\le p-1}\frac{f(i)}{i\,j\,k}\equiv
\sum... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | bc332aa7-7ba9-5e93-8e64-0d3fd16d102f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $(a_n)_{n\ge1}$ be a sequence of positive real numbers and let $s$ be a fixed positive integer. Assume that for every $n>s$
\[
a_n=\max\{a_k+a_{\,n-k}\mid 1\le k\le n-1\}\,.
\]
Prove that there exist an integer $\ell$ with $1\le\ell\le s$ and an integer $N$ such that
\[
a_n=a_\ell+a_{\,n-\ell}\qquad\text{fo... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | eb4a7a5e-fbbf-526e-aacd-c0e1abe26a57 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x, y, z$ be non‑negative real numbers. Prove that
\[
\frac{2xy^{2}}{9\,(x^{2}+y^{2})}\;\le\;\frac{y^{4}+x^{2}y^{2}+xz^{3}}{(x+y+z)^{3}}.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 202ae223-0087-509d-a19a-0c3d28dcba42 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all positive integer solutions \((x,y)\) to the equations
i) \(x^{4}-5y^{4}=1\),
ii) \(x^{4}-5y^{4}=-1\). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | e02fd5e4-4cb2-519b-b91c-27481fdd52a0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Two ants travel along the edges of a convex polyhedron. Each ant follows a closed route that starts and ends at the same vertex and does not visit any vertex (except the starting vertex) more than once (i.e. each route is a simple cycle). For every face $F$ of the polyhedron write the number of edges of $F$ belonging t... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | b1db8c0f-418b-566f-8828-3e775c4aa034 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a harmonic quadrilateral. A circle passing through $B$ and $C$ meets $AC$ again at $E$ and $AB$ again at $F$. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. Prove that the circumcircles of $\triangle ABC$, $\triangle AEF$, and $\triangle DMN$ are concurrent. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a4df2e09-62be-5634-9ce4-07f279326ff2 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let
\[
f(x)=\lim_{n\to\infty}\underbrace{x^{\,x^{\,x^{\;\cdot^{\;\cdot^{\;\cdot}}}}}}_{\text{$n$ copies of $x$}}
\]
be the infinite power tower.
1. Prove that the set of real $x$ for which this limit exists (the domain of $f$) is \((0,\,e^{1/e}]\).
2. Show that \(f\) attains its maximum value \(e\) at \(x=e^{1... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 3973dddd-b4a7-5f85-80a0-5d252987e1c4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $N$ be the set of all positive integers. Determine all functions $f\colon N\setminus\{1\}\to N$ such that for every pair of distinct integers $n,m\ge2$ the equation
$$f(n)f(m)=f\big((nm)^{2021}\big)$$
holds. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | f980028d-5aaf-5651-a656-03633ad64874 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(G\) be a simple graph with \(3n^{2}\) vertices, where \(n\ge 2\). Assume that every vertex has degree at most \(4n\), that \(G\) contains at least one vertex of degree 1, and that the distance between any two vertices of \(G\) is at most 3 (i.e. the diameter of \(G\) is \(\le3\)). Prove that the smallest possible... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 69a03fab-86e3-5b50-87e8-34dfd1d2848b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\omega(n)$ denote the number of distinct prime divisors of the positive integer $n$. Prove that
(a) there exist infinitely many ordered pairs $(a,b)$ of positive integers with $a<b$ such that
\[
\omega(a+b)=\omega(a)+\omega(b);
\]
(b) there exist infinitely many ordered pairs $(a,b)$ of positive integers wit... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 1810baec-eb08-52d5-b3f9-25f12fea04f4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
1. $f(x^{2}+f(y))=(x-y)^{2}\,f(x+y)$ for all real $x,y$;
2. $f(x+y+z)=f(x)f(1-y)+f(y)f(1-z)+f(z)f(1-x)$ for all real $x,y,z$;
3. $f(x+y)+f(x)f(y)=f(xy)+2xy+1$ for all real $x,y\,. $ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | bbb426fb-e995-5a2a-aac5-cb9f69c61560 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\omega$. Let $F_A, F_B, F_C,$ and $F_D$ be the midpoints of the arcs $AB, BC, CD,$ and $DA$ of $\omega$, respectively. Let $I_A, I_B, I_C,$ and $I_D$ be the incenters of triangles $DAB, ABC, BCD,$ and $CDA$, respectively.
Let $\omega_A$ be the circle tangent to... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | b6da05bc-f122-5278-a93d-f51be3a38b7b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Two circles $O$ and $O'$ have equal radii and intersect at two distinct points $A$ and $B$. Let $P\in O\setminus\{A,B\}$ and $Q\in O'\setminus\{A,B\}$. Choose the point $R$ so that the quadrilateral $PAQR$ is a parallelogram (with vertices taken in the order $P\!-\!A\!-\!Q\!-\!R$). Assume that the four points $B,R,P,Q$... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 00e8f1b4-9391-50a2-89e2-9a3981bd8e1f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a scalene acute‑angled triangle with incenter $I$ and circumcircle $\Gamma$. Let the line $AI$ meet $\Gamma$ again at $M$. Let $N$ be the midpoint of $BC$ and let $T$ be the point on $\Gamma$ such that $IN\perp MT$. Let the line through $I$ perpendicular to $AI$ intersect $TB$ at $P$ and $TC$ at $Q$. Prove... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | d30005f3-785c-559a-9582-d64f4fcf5163 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $p(x)=x^3-ax^2+bx-c$ be a polynomial such that $p(a)=p(b)=p(c)=0$. Find all triples of real numbers $(a,b,c)$ satisfying this condition. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 8cac421e-7daa-567a-8818-ccc2ae8e42cb | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a quadrilateral, and let $H_{1}, H_{2}, H_{3}, H_{4}$ be the orthocenters of triangles $DAB$, $ABC$, $BCD$, $CDA$, respectively. Prove that the area of quadrilateral $ABCD$ equals the area of quadrilateral $H_{1}H_{2}H_{3}H_{4}$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a1c06657-880c-5315-b65b-eb2b180bd26d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\mathbb{R}^{+}$ denote the set of positive real numbers. Find all functions $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ such that for all $x,y>0$
\[
f\!\big(2022\,x\,f(x)+f(y)\big)=y+2022\,f(x)^{2}.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 65095b86-72a8-51f0-89fb-639925bb5a98 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with incenter $I$ and circumcenter $O$. Let $\omega$ be a circle tangent to sides $AB$ and $AC$ and tangent to the circumcircle $(ABC)$. Let $T$ be the common point of $\omega$ and $(ABC)$. Let line $IO$ intersect $AT$ at $K$. Prove that $IK \le IO$, and determine when equality occurs. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | b710dff6-5e4b-5186-8878-f8f41dc2ce4a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a convex quadrilateral such that $\angle ABC=\angle ADC=135^\circ$.
Let $M$ and $N$ be points on the rays $AB$ and $AD$, respectively, with $\angle MCD=\angle NCB=90^\circ$.
The circumcircles of $\triangle AMN$ and $\triangle ABD$ intersect again at $K\neq A$.
Prove that $\displaystyle\angle AKC=90^... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 8f477fe1-316d-5217-becb-1ac80aec383b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with circumradius $R$ and inradius $r$. Prove that for every point $P$ in the plane of $ABC$,
$$PA+PB+PC \;\ge\; 2\sqrt{\,r\bigl(4R+r\bigr)}\,.$$ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 7b74a5a6-b028-5c5b-94b9-87d5fe06e096 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be the side lengths of a triangle. Prove that
\[
\frac{(a+b)(b+c)(c+a)}{8}\ge \frac{(2a+b)(2b+c)(2c+a)}{27}.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 6787c868-5fd8-55eb-859f-6a81456e48c1 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ with $a+b+c=3$. Prove that
\[
2\bigl(a^{4}+b^{4}+c^{4}\bigr)+\frac{4ab}{(a+b)^{2}}+\frac{4bc}{(b+c)^{2}}+\frac{4ca}{(c+a)^{2}}
\;\ge\;3\bigl(a^{2}+b^{2}+c^{2}\bigr).
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 7ddd46fb-2392-5c65-998f-e3e86b49e6e7 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(a,b,c\ge 0\) satisfy \(a+b+c=3\). Prove that
\[
\frac{1}{2+a^{5}b^{5}}+\frac{1}{2+b^{5}c^{5}}+\frac{1}{2+c^{5}a^{5}}\ge 1.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 52a5f73d-7cbb-5e09-93ff-3484180ff3d7 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $P$ be an interior point of $\triangle ABC$. Let $D$, $E$, $F$ be the orthogonal projections of $P$ onto $BC$, $CA$, and $AB$, respectively. Let $S$ be the midpoint of side $AB$. Prove that the line $FP$ bisects $\angle DFE$ of $\triangle DEF$ if and only if $SD = SE$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 513837f2-1320-5581-8709-c8af443af4ee | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Suppose a function $f:\mathbb{R}\to\mathbb{R}$ satisfies $f(f(x))=-x$ for every real $x$. Prove that $f$ has infinitely many points of discontinuity. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | d76afb4c-1e2f-5283-ae20-6473a44e5682 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $X:\mathbb{R}^2\to\mathbb{R}^2$ be a bijection such that for every (Euclidean) circle $C$ in the plane the image $X(C)$ is also a circle. Prove that $X$ maps every straight line onto a straight line. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | e762a70a-1ff3-5c0e-8c51-980a73585600 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n$, $m$, and $k$ be positive integers satisfying $(n-1)n(n+1)=m^k$. Prove that $k=1$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 0ccd3d9c-09e9-5721-b251-162d5d4b92a1 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a$ be a fixed positive integer. For a prime $p$ with $p\nmid a$, denote by $\operatorname{ord}_p(a)$ the smallest positive integer $k$ such that $a^{k}\equiv1\pmod p$. Prove that the ratio
\[
\frac{p-1}{\operatorname{ord}_p(a)}
\]
is not bounded as $p$ runs over all primes; equivalently, for every $N\in\mathbb N$... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | fba7665b-9c82-5d03-80f6-6fee94233791 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ such that $\displaystyle\frac1a+\frac1b+\frac1c=1$. Prove that
\[
\frac{\sqrt a}{a^{2}+3bc}+\frac{\sqrt b}{b^{2}+3ac}+\frac{\sqrt c}{c^{2}+3ab}\le \frac{3}{4\bigl(\sqrt a+\sqrt b+\sqrt c\bigr)}\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 45d03c3e-5d94-55c0-88ed-23efe2e61be6 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with circumcircle $(O)$. Let $d$ be an arbitrary line and let $d_1$, $d_2$, $d_3$ be the reflections of $d$ across the sides $BC$, $CA$, $AB$, respectively. Define $A_1 = d_2\cap d_3$, $B_1 = d_3\cap d_1$, $C_1 = d_1\cap d_2$. Prove that the incenter of triangle $A_1B_1C_1$ lies on the circumcir... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | f33c9543-e71b-5d9a-ae3e-4686f01f1396 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a cyclic quadrilateral. Let $M$ be its Miquel point and let $P=AC\cap BD$. Reflect $M$ across the lines $AB$, $BC$, $CD$, $DA$ to obtain points $M_1$, $M_2$, $M_3$, $M_4$, respectively. Prove that $M_1$, $M_2$, $M_3$, $M_4$, and $P$ are collinear. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 46b75d04-0083-540c-9f25-bcfa05e6ae3f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $A', B', C'$ be points on the sides $BC$, $CA$, and $AB$ of triangle $ABC$, respectively, such that the cevians $AA', BB', CC'$ are concurrent at a point $P$. For each of $A', B', C'$, let $\Omega_A$, $\Omega_B$, $\Omega_C$ be the circles centered at $A'$, $B'$, $C'$ that pass through $P$ (respectively). Define $A'... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 71ce3437-9860-528c-8d4e-ce7aa0719ff0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In triangle $ABC$, let $AD$ be the altitude from $A$ to $BC$. Let $I$ be the incenter of $ABC$, and let $P$ be the midpoint of side $BC$. Let $M$ be the midpoint of the minor arc $BC$ of the circumcircle of $ABC$. Let $X$ be the point where the incircle of $ABC$ meets $AD$, and assume that $X$, $I$, and $P$ are colline... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 397e1878-c339-5a27-aa8b-eb23354e0492 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In triangle $ABC$, let $D\in AC$, $E\in BC$, and $F\in AB$ such that the cevians $BD$, $CF$, and $AE$ are concurrent at a point $Q$. Let $P$ be the foot of the perpendicular from $Q$ to $DF$, i.e. $AQ\perp DF$ at $P$. Suppose the quadrilateral $BPDC$ is cyclic. Prove that $BD = CD\,. $ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 454fa0b7-7a2f-54c8-922a-3632e392f7bc | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Solve for integers \(x\) and \(y\): \(x^{2}=y^{3}+13\). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 04c807d4-afe5-567e-b190-566cb4fddcc0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle inscribed in a circle $\Gamma$. Let $K$ and $L$ be points on sides $CA$ and $AB$, respectively. The tangent to $\Gamma$ at $B$ meets $CL$ at $E$, and the tangent to $\Gamma$ at $C$ meets $BK$ at $F$. Let $D=BK\cap CL$. Let $I$ and $J$ be the centers of the circumcircles of $\triangle BDL$ and $\... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | c1a634ab-66b4-59b4-95e7-b863a562814a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}_+\to\mathbb{R}_+$ such that for every three positive real numbers $x,y,z$ the following holds:
$$f(x+y+f(y)) + f(x+z+f(z)) = f(2f(z)) + f(2x) + f(2y).$$ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | bb531a79-533d-542b-971a-117cd8dd9133 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with circumcircle $(O)$ and orthocenter $H$. Let $M$ be the midpoint of $BC$. Define $D = AH\cap BC$, let $N\neq A$ be the second intersection of line $AM$ with $(O)$, and let $P\neq D$ be the second intersection of the circle through $A$, $D$, $N$ with line $BC$. Prove that $\angle OPA = \angle... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | eaef55f3-5ce6-50f1-8e97-30f78c31ef8b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine all functions \(f:\mathbb{R}^{+}\to\mathbb{R}^{+}\) such that
\[
f\bigl(f(x)+x\,f(y)\bigr)=x+y\,f(x)
\qquad\text{for all }x,y>0.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | d666b9c8-7dc7-51a6-86cd-187236042187 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Does there exist two infinite subsets $S$ and $T$ of the positive integers such that every positive integer $n$ can be written uniquely in the form
$$n = s_1t_1 + s_2t_2 + \dots + s_k t_k,$$
where $k\ge 1$ depends on $n$, the $s_i$ are distinct elements of $S$ satisfying $s_1 < s_2 < \dots < s_k$, and each $t_i$ belo... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | bda0764c-b138-5ffd-a768-ab7c21e3de9e | [] | AoPS | Nemotron-Math-Proofs-v3 |
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