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"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying
\[f(x+y)+f(x)f(y)=f(xy)+f(x)+f(y)\qquad\forall\,x,y\in\mathbb{R}.\] | {
"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 | e6ae8a18-ab09-588f-abb1-02ec62435c4f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | A circle ω is inscribed in triangle ABC, tangent to side BC at point K. Let ω′ be the image of ω under a half‑turn (central symmetry) about point A. Let A₀ be the intersection of the two tangents drawn from B and from C to ω′ (i.e., BA₀ and CA₀ are tangent to ω′). Let M be the midpoint of side BC. Prove that line AM pa... | {
"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 | 6ad5d9c1-c6e3-5c04-bfe0-311f9dbe0427 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all pairs of positive integers $(a,b)$ such that
$$a^4 = 2b^2 - 1.$$ | {
"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 | 70377641-0797-56e7-88e4-8c6b30cc2cba | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let the Fibonacci sequence be defined by $f_1 = f_2 = 1$ and $f_{n+2}=f_{n+1}+f_n$ for $n\ge1$. Determine the number of positive integers $n$ for which $\sqrt{f_n}\in\mathbb Z$ (i.e., $f_n$ 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 | daca4187-2c92-572d-a499-501f5529e0d6 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Consider an acute‑angled triangle $ABC$ with $AB=AC$ and $\angle A>60^\circ$. Let $O$ be the circumcenter of $ABC$. Let $P$ be a point on the circumcircle of $\triangle BOC$ such that $BP\parallel AC$. Let $K$ be a point on the segment $AP$ satisfying $BK=BC$. Prove that $CK$ bisects the arc $BC$ of the circumcircle of... | {
"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 | 5b536e5e-8942-5af8-ad86-7fee118a4dc1 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with inradius $r$ and circumradius $R$. Denote by $O$ its circumcenter and by $H$ its orthocenter. Let $P$ be a point on the segment $OH$ that lies inside $\triangle ABC$. Prove that
$$6r\;\le\;PA+PB+PC\;\le\;3R\,.$$ | {
"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 | 26bf784b-c725-5204-9376-a13e1dd0f249 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,y,z>0$ with $x+y+z=1$. Prove that
\[
\frac{xy}{\sqrt{xy+yz}}+\frac{yz}{\sqrt{yz+zx}}+\frac{zx}{\sqrt{zx+xy}}
\le \frac{3}{4}\sqrt{3(x+y)(y+z)(z+x)}\, .
\] | {
"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 | a89cdf67-310c-5de1-8337-1be3c490df0f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb R\to\mathbb R$ such that for all real numbers $x,y$,
$$f\bigl(f(x)-f(y)\bigr)=(x-y)^2\,f(x+y).$$ | {
"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 | 9711ea86-2d28-5b5f-a23f-2c254a092abe | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ satisfy $a^2+b^2+c^2=3$. Prove that
\[
\frac{a}{b^2+c^2+7bc+3}+\frac{b}{c^2+a^2+7ca+3}+\frac{c}{a^2+b^2+7ab+3}\le\frac{3}{4abc\,(ab+bc+ca)}.
\] | {
"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 | 114efe23-1265-54dd-9c44-a4531847b29c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For positive real numbers $x$, $y$, $z$, prove that
\[
\frac{(x+y+z)^{3}}{xyz}+18\sum_{\text{cyc}}\frac{x^{2}}{x^{2}+yz}\ge 54\,.
\] | {
"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 | f0b31635-0dbf-55a1-b91c-7fbd5115f840 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all integers \(n\ge 1\) such that \(2^{n}-1\) has exactly \(n\) positive integer divisors. | {
"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 | 02e9fce0-6aec-543e-ad5a-e9802642522a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
\[
f\bigl(x+f(x+y)\bigr)=f(x-y)+f\bigl(f(x)\bigr)
\]
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 | d5b88665-a17d-5270-a3d7-ab3cc436f308 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a$, $b$, and $c$ be positive real numbers. Prove that
\[
\frac{a^{3}}{(b+c)^{2}}+\frac{b^{3}}{(c+a)^{2}}+\frac{c^{3}}{(a+b)^{2}}
\;\ge\;
\frac{11\bigl(a^{2}+b^{2}+c^{2}\bigr)-8\bigl(ab+bc+ca\bigr)}{4\,(a+b+c)}.
\]
Equality should be characterized. | {
"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 | 51d391f2-d579-57c8-a265-d2efd8ad7993 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine whether there exists a collection of 23 real numbers (repetitions allowed, each occurrence counted as a distinct element) such that exactly 2422 of its non‑empty subsets have a rational product. If such a collection exists, provide an example. | {
"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 | f23373b7-68ea-5e47-be96-0f60178267a8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute triangle. Let $H_a$ and $H_b$ be the feet of the altitudes from $A$ and $B$ onto $BC$ and $AC$, respectively. Let $D_a$ and $D_b$ be the points where the internal angle bisectors from $A$ and $B$ meet $BC$ and $AC$, respectively. Let $A'$ and $B'$ be the points where the incircle of $ABC$ touches ... | {
"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 | 2584bb7c-573d-53b8-ba98-30bf44885955 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Suppose a finite collection of polygons (convex or non‑convex) in the plane is such that any two of them have at least one point in common. Prove that for every point $A$ in the plane
a) there exists a straight line passing through $A$ that meets every polygon, and
b) there exists a circle with centre $A$ that me... | {
"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 | 6ef54792-412d-5ed3-ac4f-c77bee952ba7 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all positive integers $m,n\;(m,n\ge 2)$ such that for every $a\in\{1,2,\dots ,n\}$ one has
$$a^{\,n}\equiv 1\pmod m.$$ | {
"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 | 3650dd15-002a-585b-9af7-e77663f6825e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Consider a cyclic quadrilateral $ABCD$. Let $E=AB\cap CD$. Let $M$ be the second intersection point of the circumcircles of $EBC$ and $EDA$, i.e. $(EBC)\cap(EDA)=\{E,M\}$. Let $N=AC\cap BD$. Let $K$ be the second intersection of the line $EN$ with the circle $(EBC)$, i.e. $EN\cap(EBC)=\{E,K\}$. Prove that the line $MK$... | {
"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 | a9b20153-ceff-5b75-8075-cdd977ca2bfa | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute scalene triangle with circumcenter $O$, and let $T$ be the point on line $BC$ such that $\angle TAO=90^{\circ}$. The circle with diameter $\overline{AT}$ meets the circumcircle of $\triangle BOC$ at two points $A_{1}$ and $A_{2}$, where $OA_{1}<OA_{2}$. Define $B_{1},B_{2}$ and $C_{1},C_{2}$ analo... | {
"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 | f96d5dc9-b547-5041-a8d5-312af47b8a60 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Within a group of $2009$ people, any two distinct people have exactly one common friend. Find the smallest possible value of the difference between the number of friends of the person with the most friends and the number of friends of the person with the fewest friends. | {
"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 | fbaf09b0-15f0-5433-89b3-2f8b31dbe627 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\in\mathbb{R}$. Prove that
\[
\bigl(a^{3}+b^{3}+c^{3}-3abc\bigr)^{2}
+\frac92\,(a+b+c)\,(ab^{4}+bc^{4}+ca^{4})
\ge 18a^{2}b^{2}c^{2}
+\frac{15}{2}\,(a^{3}b^{3}+b^{3}c^{3}+c^{3}a^{3}).
\]
Determine all triples $(a,b,c)$ for which equality 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 | b163c732-74b1-5aed-a661-2bb0916782f4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Convex quadrilateral $ABCD$ is cyclic with circumcircle $w$. Let the extensions of sides $AB$ and $DC$ intersect at $K$. Choose a point $L$ on the diagonal $BD$ such that $\angle BAC=\angle DAL$. Let $M$ be the point on segment $KL$ satisfying $CM\parallel BD$. Prove that the line $BM$ is tangent to $w$ (i.e., $BM$ tou... | {
"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 | 41b5e993-c104-596c-ab0e-382243ed07ed | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions \(f:\mathbb{R}\to\mathbb{R}\) such that
\[
f(x)\,f\!\bigl(yf(x)-1\bigr)=x^{2}f(y)-f(x)
\]
for all real numbers \(x,y\). | {
"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 | d2254b21-73b5-554d-bb32-e15677bfd1a5 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\omega_1$ be a circle with centre $O$ and let $\omega_2$ be another circle that passes through $O$ and intersects $\omega_1$ at points $A$ and $B$. A line $\ell$ through $O$ meets $\omega_1$ again at $C$ and $Z$ (so $CZ$ is a diameter of $\omega_1$) and meets $\omega_2$ again at $D$ (besides $O$). Prove that $D$ i... | {
"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 | 5107cb68-525e-5376-a4ae-e68b74cdf3ec | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In triangle $ABC$, let $BB'$ and $CC'$ be the internal angle bisectors, meeting sides $AC$ and $AB$ at $B'$ and $C'$ respectively. Their intersection is the incenter $I$. Let the line $B'C'$ intersect the circumcircle of $ABC$ again at points $M$ and $N$ (the two intersection points of $B'C'$ with the circumcircle). If... | {
"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 | 9bcdd779-6b30-5127-9ec6-e439eba630a8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle and $P$ a point. Denote by $A_1B_1C_1$ the pedal triangle of $P$ with respect to $ABC$, and let $P^*$ be the isogonal conjugate of $P$ (with respect to $ABC$). Let $A_2B_2C_2$ be the pedal triangle of $P^*$. Choose any point $Q$ on the line $PP^*$. The lines $A_2Q$, $B_2Q$, $C_2Q$ meet the circu... | {
"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 | 76ef9dea-aef4-5cb5-ae8d-600a843a23b3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given $a,b,c>0$, prove that
\[
\frac{1}{a\sqrt{8a+b}}+\frac{1}{b\sqrt{8b+c}}+\frac{1}{c\sqrt{8c+a}}\ge \frac{1}{\sqrt{abc}}.
\] | {
"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 | 6c68aa31-7ed6-5e1a-a2fc-d2cd65119e00 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Maker and Breaker are building a wall. There are $m$ columns (ground squares) labeled $1,2,\dots,m$, and each column may contain at most $n$ blocks stacked from the ground upward. The players take turns placing one block on a column; a block must be placed either on the ground (height $1$) or on top of the highest exis... | {
"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 | 23718f17-1973-5738-8099-aea992398d9d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle. Let $X$ be the $A$‑excenter and $I$ the incenter of $ABC$. Let $M$ be the circumcenter of $\triangle BIC$, and let $G$ be the point on $BC$ such that $XG\perp BC$. Construct the circle $\Omega$ with diameter $AX$. If $\Omega$ and the circumcircle of $\triangle ABC$ intersect again at a point $P... | {
"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 | d41f4076-53c0-5f57-9320-025cba241789 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | If $a,b,c>0$, prove
\[
\text{(i)}\qquad \frac{(a+b)(b+c)(c+a)}{8abc}\ge \frac12+\frac{a^2+2b^2+c^2}{2(a+b)(b+c)},
\]
\[
\text{(ii)}\qquad \frac{3(a+b)^2(b+c)^2}{4ab^2c}\ge 7+\frac{5(a^2+2b^2+c^2)}{(a+b)(b+c)}.
\] | {
"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 | 18290e38-e340-58b1-9255-139ad1ed9c91 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | A 9 × 2004 rectangular array is to be filled with the integers 1,2,…,2004, each integer appearing exactly nine times in the whole array. The entries must be placed so that in every column the difference between the largest and the smallest entry is at most 3. Determine the smallest possible sum of the numbers in the fi... | {
"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 | e2fe5f76-e00c-52df-b571-81844496b790 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,\;y,\;z\ge 0$ satisfy $x^{2}+y^{2}+z^{2}+xyz=4$. Prove that
\[
\sqrt{x+y}\;+\;\sqrt[3]{\,y+z\,}\;+\;\sqrt[4]{\,z+x\,}<4\,.
\] | {
"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 | 40364125-c95c-50fb-84c3-088d534284f5 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying
\[
f\bigl(f(x)f(y)-f(x)+y\bigr)=f(xy)+2y\qquad\forall x,y\in\mathbb{R},
\]
and find all functions $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ satisfying
\[
f\bigl(xf(y)+2022\bigr)=y\,f(x+y+2022)\qquad\forall x,y>0.
\] | {
"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 | ce772516-66f9-5580-90ac-9012147ec59f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCDEF$ be a convex hexagon. Denote by $AB,BC,CD,DE,EF,FA$ its side lengths and by $AD,BE,CF$ the lengths of the three main diagonals. Suppose
\[
AD = BC + EF,\qquad BE = AF + CD,\qquad CF = DE + AB.
\]
Prove that
\[
\frac{AB}{DE}= \frac{CD}{AF}= \frac{EF}{BC}\, .
\] | {
"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 | e1b2f42c-6124-501d-923f-2c21ee0d37fd | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that for any integer $n\ge 0$ and any collection of $2n+1$ irrational real numbers, one can choose $n+1$ of them such that the sum of any non‑empty subcollection of the chosen $n+1$ numbers is irrational. In other words, there exists a subset $A$ of size $n+1$ with the property that for every $k\in\{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 | bfbb3480-1da8-5635-a314-9851b1f781d8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(P(x,y,z)\) be a polynomial in three real variables with real coefficients satisfying the identities
\[
P(x,y,z)=P(x,y,xy-z)=P(x,zx-y,z)=P(yz-x,y,z)
\qquad\text{for all real }x,y,z.
\]
Prove that there exists a univariate polynomial \(F(t)\) with real coefficients such that
\[
P(x,y,z)=F\bigl(x^{2}+y^{2}+z^{... | {
"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 | 0a6355d4-be49-5c90-95d3-a66bbfe30b2e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b>0$ satisfy $a^5+b^5=\frac{1}{5}$. Prove that
\[
\Bigl(8+\frac{3}{5ab}\Bigr)(a+b)\ge 11,
\]
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 | 9a38052b-12ce-563a-80a2-50c270aac1b2 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | A circle meets the three sides $BC$, $CA$, $AB$ of triangle $ABC$ at points $D_1,D_2$, $E_1,E_2$, $F_1,F_2$, respectively. Let $L = D_1E_1\cap D_2F_2$, $M = E_1F_1\cap E_2D_2$, $N = F_1D_1\cap F_2E_2$. Prove that the lines $AL$, $BM$, $CN$ 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 | 1b16ed04-1965-55c1-9e64-1b4b2b9f28ab | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute triangle with circumcenter $O$ and incenter $I$. Choose points $E\in AC$ and $F\in AB$ such that $BF=CE=BC$. Let $(AEF)$ denote the circumcircle of $\triangle AEF$, and let $G\neq A$ be its second intersection with the circumcircle $(ABC)$. Let $J$ be the circumcenter of $\triangle AEF$.
a) 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 | 846bad64-0e7b-562c-bca9-41441040217d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n$, $k$, and $m$ be positive integers with $n>2k$. Let $S$ be a non‑empty collection of $k$‑element subsets of $\{1,2,\dots,n\}$ such that every $(k+1)$‑element subset of $\{1,2,\dots,n\}$ contains exactly $m$ members of $S$. Prove that $S$ must contain every $k$‑element subset of $\{1,2,\dots,n\}$; i.e. $S=\{A\su... | {
"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 | 2ea391ac-a394-52fb-b9da-569f1661f3ad | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Ten million fireflies are placed in ℝ³. Friendship is a mutual relation between fireflies. Each second exactly one firefly moves to a new position such that its distances to each of its friends remain unchanged; this is the only way the fireflies ever change position, and no two fireflies ever occupy the same point. In... | {
"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 | 9c6de9fa-0c01-56d4-9bcb-b1ca9d9d7051 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all natural numbers $n$ for which there exists a permutation $\sigma$ of $\{1,2,\ldots,n\}$ satisfying
\[
\sum_{i=1}^{n}\sigma(i)(-2)^{\,i-1}=0.
\] | {
"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 | abb6b0d8-e482-536f-9a76-750407364883 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given a polygon with vertices labeled cyclically, each of its n sides may be taken in either orientation (e.g., side AB or BA). Form a linear list containing each of the 2n directed sides exactly once, with the requirement that for any two consecutive items in the list the ending vertex of the first item is different 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 | 7aa0db6a-455a-5bb7-8043-951e4e7132f0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b\ge0$. Prove that $9a^{4}+8a^{3}b+7a^{2}b^{2}-24ab^{3}+9b^{4}\ge0$. | {
"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 | 15f7f44c-ff73-535e-a8ba-150d32b0b359 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In triangle $ABC$ with incenter $I$, let the incircle and the $A$‑excircle touch side $BC$ at $A'$ and $A''$, respectively. Let $\odot(A'A'')$ denote the circle with diameter $A'A''$; its intersection with line $AI$ consists of two points $A_1$ and $A_2$. Let $A_3$ be the foot of the altitude from $A$ to $BC$ (so $AA_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 | a3274b5a-c444-5b6a-a8c2-267ab09403a4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $f:\mathbb{N}\to\mathbb{N}$ satisfy
$$k\,f(n)\le f(k n)\le k\,f(n)+k-1$$
for all positive integers $k,n$.
(a) Prove that for all $a,b\in\mathbb{N}$
$$f(a)+f(b)\le f(a+b)\le f(a)+f(b)+1.$$
(b) Suppose in addition that $f$ satisfies
$$f(2007\,n)\le 2007\,f(n)+2005$$
for every $n\in\mathbb{N}$. 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 | 86892eb1-a82b-5e28-bc78-a442b50ed48f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(a,\;b,\;c\) be non‑negative real numbers. Prove that
\[
a^{2}+b^{2}+c^{2}+ab+bc+ca\;\ge\;a\sqrt{3b^{2}+c^{2}}+b\sqrt{3c^{2}+a^{2}}+c\sqrt{3a^{2}+b^{2}}.
\]
(Here \(\sum a^{2}=a^{2}+b^{2}+c^{2}\) and \(\sum ab=ab+bc+ca\).) | {
"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 | e9e3c6ee-5f6b-579e-8cf2-d2ef1b36ef24 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine whether there exists an integer $n$ such that the decimal representation of $2^n$ begins with the four‑digit string “2007”. If such an $n$ exists, give the smallest positive $n$ (i.e. the least $n$ for which the leading digits of $2^n$ are 2007). | {
"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 | 2c270422-8912-57e5-9c13-e10f6d4802f1 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Suppose $a,b,c$ are positive real numbers satisfying $a+b+c=3$. Prove that
\[
\frac{1}{2+a^{2}+b^{2}}+\frac{1}{2+b^{2}+c^{2}}+\frac{1}{2+c^{2}+a^{2}}\le \frac34 .
\]
Equality occurs for $a=b=c=1$. | {
"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 | 8fbb2b1f-a384-537c-90c2-8bc199fdae0d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n$ be a fixed positive integer. Form the sequence $s$ by concatenating all $n!$ permutations of $\{1,2,\dots,n\}$ written in lexicographic order; thus $s$ is a sequence of $n\cdot n!$ integers, each entry belonging to $\{1,\dots,n\}$. Prove that if any $n$ consecutive terms of $s$ have sum $\displaystyle\frac{n(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 | bb5980e4-db72-55c9-a924-975c669f54a2 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a trapezoid with $AB\parallel CD$. Let $P$ be the intersection of the diagonals $AC$ and $BD$. Let $Y$ be a point on $BC$ such that $\angle AYB = \angle DYC$, and let $X$ be a point on $AD$ such that $\angle AXB = \angle DXC$. Prove that $PX = PY$. | {
"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 | 4aed4fd2-3aaf-5499-9d8a-50c39d4ae245 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,y,z$ be real numbers satisfying $0<x<y<z<\dfrac{\pi}{2}$. Prove that
\[
\sqrt{2}+2\sin x\cos y+2\sin y\cos z>\sin2x+\sin2y+\sin2z\,.
\] | {
"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 | 343291f6-9f3f-563d-aa7e-996b738c2666 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with incenter $I$ and circumcenter $O$. Choose points $A_1$, $B_1$, $C_1$ on the segments $AI$, $BI$, $CI$, respectively. Let $A_2$, $B_2$, $C_2$ be the vertices of the triangle formed by the three perpendicular bisectors of the segments $AA_1$, $BB_1$, $CC_1$ (so each $A_2$, $B_2$, $C_2$ is 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 | 61e263a7-3cf2-532b-ae8c-09ef09ffa133 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find the maximal value of
\[
S=\sqrt[3]{\frac{a}{b+7}}+\sqrt[3]{\frac{b}{c+7}}+\sqrt[3]{\frac{c}{d+7}}+\sqrt[3]{\frac{d}{a+7}},
\]
where $a,b,c,d\ge0$ and $a+b+c+d=100\,. | {
"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 | 8c5104c9-8839-5dbc-b014-e21ff2a24ef3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $(C)$ be the circumcircle of triangle $ABC$. The tangents to $(C)$ at $B$ and at $C$ intersect at $D$. Let $I$ be the midpoint of $BC$, and let $AI$ intersect $(C)$ again at $K$. Prove that $\displaystyle\angle DAB=\angle IAC\,$. | {
"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 | 94c3f304-7fc9-5961-ad4a-9bb5d92905e8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCDEF$ be a convex hexagon such that
\[
AB=BC=CD,\qquad DE=EF=FA,
\]
and
\[
\angle BCD=\angle EFA=\frac{\pi}{3}.
\]
Let $G$ and $H$ be interior points of the hexagon satisfying
\[
\angle AGB=\angle DHE=\frac{2\pi}{3}.
\]
Prove that
\[
AG+GB+GH+DH+HE\ge CF.
\] | {
"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 | 505a366a-c7f0-536c-9a1e-3c027001cc8e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $A$ be the complete graph $K_n$ on $n\ge3$ vertices. In each turn you may delete any one edge that belongs to a (simple) cycle of length $4$ in the current graph. What is the maximal possible number of turns (edge deletions) that can be performed? | {
"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 | d9fbc025-4f78-53f4-8518-a7ff058ad31e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with circumcircle $(O)$. Let $A^{+}$ be the midpoint of the arc $BAC$ of $(O)$. Let $D$ be the foot of the internal angle bisector of $\angle BAC$ on side $BC$. Let $O_{c}$ be the circumcenter of triangle $CAD$. Prove that the points $A^{+}$, $C$, and $O_{c}$ 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 | 32844a48-7d9d-5434-94d7-72935d41cc98 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Initially, each unit square of an $n\times n$ grid is colored red, yellow, or blue. In each round the following operation is performed simultaneously for every unit square:
- If a square is red and has a neighboring (edge‑adjacent) yellow square, it becomes yellow.
- If a square is yellow and has a neighboring blue squ... | {
"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 | 3fd831ee-5098-5aa7-914b-f0be4df3dabb | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle and $(O)$ its circumcircle.
Let the perpendicular bisector of $BC$ meet $CA$ at $A_2$ and $AB$ at $A_1$.
Define the other points cyclically: the perpendicular bisector of $CA$ meets $AB$ at $B_2$ and $BC$ at $B_1$, and the perpendicular bisector of $AB$ meets $BC$ at $C_2$ and $CA$ at $C_1$.... | {
"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 | 56183be1-90a4-5830-b24e-e34021d8750b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f,g:\mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$,
$$f(x^{3}+2y)+f(x+y)=g(x+2y).$$ | {
"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 | 4eabc38a-f32e-55e1-b395-3f5a1d42a37c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Solve each of the following systems of equations for real numbers \(x\) and \(y\):
1. \(\displaystyle \begin{cases}
x^{4}-y^{4}=240\\[2pt]
x^{3}-2y^{3}=3\bigl(x^{2}-4y^{2}\bigr)-4\bigl(x-8y\bigr)
\end{cases}\)
2. \(\displaystyle \begin{cases}
x^{4}+x^{3}y+9y=y^{3}x+y^{2}x^{2}+9x\\[2pt]
x\bigl(y^{3}-x^{3}\bigr)=7
\end... | {
"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 | 55fe2b99-65d9-5cbd-ab6f-e9af6b830b6d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $D$ be a point in the plane of triangle $ABC$.
Define
* $E$ as the isogonal conjugate of $A$ with respect to $\triangle BDC$,
* $F$ as the isogonal conjugate of $C$ with respect to $\triangle ABD$, and
* $G$ as the isogonal conjugate of $B$ with respect to $\triangle ADC$.
Prove that the circumcircles of... | {
"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 | 82f75de9-5dbb-5b44-bb9f-3d748e5560b9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle. Construct triangles $PAB$ and $QAC$ outside of $ABC$ such that $AP=AB$, $AQ=AC$, and $\angle BAP=\angle CAQ$. Let $BQ$ and $CP$ intersect at $R$, and let $O$ be the circumcenter of $\triangle BCR$. Prove that $AO\perp PQ$. | {
"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 | 55d7f865-eaf3-521a-8e3b-c338959df51c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle. Let $P$ be a PC‑point of $ABC$, let $Q$ be the Ceva point of $P$, let $S$ be the isotomcomplement of $Q$, let $S^*$ be the isogonal conjugate of $S$, and let $Q^*$ be the isogonal conjugate of $Q$. Let $O$ denote the circumcenter of $ABC$. Prove that $S^*$ lies on the line $OQ^*P$ (i.e. $O,\;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 | c21a92e3-2da7-5faf-80b0-76aef0281a8c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that for a tetrahedron $ABCD$, the sum of the three face angles meeting at each vertex equals a straight angle (π) if and only if the four faces are congruent triangles. | {
"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 | e9924362-d7a3-5bfd-860e-e199d205e6fc | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $f(n)$ denote the number of $n$‑tuples $(a_1,\dots,a_n)$ of integers with $a_i\ge 2$ such that in the extended sequence $1,a_1,\dots,a_n,1$ each interior term divides the sum of its two neighbours; i.e. for $1\le i\le n$ we have $a_i\mid(a_{i-1}+a_{i+1})$, where $a_0=a_{n+1}=1$. Show that $f(n)$ equals the $n$‑th 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 | 4a184cf0-10fb-51a1-86eb-4231339c475f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given non‑negative real numbers $a,b,c$ such that $a+b+c=3$, prove that
\[
\sqrt{a+bc}+\sqrt{b+ca}+\sqrt{c+ab}\;\ge\;\sqrt2\,(ab+bc+ca)\,.
\] | {
"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 | 0d6206dc-e3d3-55d9-9d95-faaa5e9dafa3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Do there exist two polynomials with integer coefficients such that each polynomial has a coefficient with absolute value exceeding $2015$, while all coefficients of their product have absolute values not exceeding $1$? | {
"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 | d9ca7f14-4b00-54aa-88ea-b6af78127f98 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be an isosceles trapezoid with $AB\parallel CD$, $AD=BC$, and $AB>CD$.
Let $E$ be a point such that $EC=AC$, $EC\perp BC$, and $\angle ACE<90^\circ$.
Let $\Gamma$ be the circle with centre $D$ and radius $DA$, and let $\Omega$ be the circumcircle of $\triangle AEB$.
Suppose $\Gamma$ and $\Omega$ 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 | 0de620a5-deb8-52df-bfea-003203a4afaf | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:[0,\infty)\to[0,\infty)$ such that
\[
f(x^2)+f(y)=f\!\bigl(x^2+y+x\,f(4y)\bigr)
\]
for every $x,y\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 | a0cf6cc2-9e8b-5adb-bcf6-0d90e904cb5e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | If $a,b,c,d>0$ and $abcd=1$, prove that
\[
\frac{ab+1}{a+1}+\frac{bc+1}{b+1}+\frac{cd+1}{c+1}+\frac{da+1}{d+1}\ge 4,
\]
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 | d8d5d068-f014-5819-b319-948373a01b9b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ such that
$$f(x+f(xy)) = x\,f\bigl(1+f(y)\bigr)$$
for all positive 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 | a3a2ab23-7673-54e5-b991-faabdf47e85c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c,d$ be positive real numbers. Prove that
\[
\frac{1}{a(1+b)}+\frac{1}{b(1+c)}+\frac{1}{c(1+d)}+\frac{1}{d(1+a)}\ge \frac{16}{1+8\sqrt{abcd}}\,.
\] | {
"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 | 74254f2b-3123-543f-b92f-3c332120e112 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given an integer $k\ge 2$, determine the largest integer $n$ for which there exists a graph $G$ with metric dimension $\dim(G)\le k$ that contains a subgraph isomorphic to the wheel $W_n$ (the wheel formed by a cycle $C_n$ together with a universal hub vertex). | {
"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 | d1aec470-80ef-59c6-83a9-c4ae240a5d6d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle. Let $(1)$ be the incircle of $ABC$ with centre $I$. Let $D,E,F$ be the points where $(1)$ touches $BC,CA,AB$ respectively (the contact triangle). Let $A'$ be the midpoint of $BC$. Let $V = BI\cap EF$ and $W = CI\cap EF$. Let $Fe$ denote the Feuerbach point of $ABC$ (the point where the incircle... | {
"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 | e3fcaa32-ba6f-52d0-a451-58acee7854c9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ satisfy $abc=1$. Find the minimum value of
$$P=\frac{1}{a\sqrt{a+b}}+\frac{1}{b\sqrt{b+c}}+\frac{1}{c\sqrt{c+a}}\,. $$ | {
"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 | 486fea63-9270-5009-8617-9e9ac0efbad7 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $r_1,r_2,\dots,r_n\in\mathbb{R}$ be given, where $n$ is a positive integer. Prove that there exists a subset $I\subseteq\{1,2,\dots,n\}$ such that for every $i=1,2,\dots,n-2$ we have
\[
1\le |I\cap\{i,i+1,i+2\}|\le 2,
\]
and
\[
\Bigl|\sum_{i\in I} r_i\Bigr|\;\ge\;\frac16\sum_{i=1}^{n}|r_i|.
\] | {
"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 | 183c1889-3321-5d97-9c9a-43523d77a7e9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ satisfy $abc=1$. Prove that
\[
\frac{1}{(a+1)^2(b+c)}+\frac{1}{(b+1)^2(c+a)}+\frac{1}{(c+1)^2(a+b)}\le \frac{3}{8}.
\] | {
"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 | ee752652-f3d8-5fd3-9285-77a65b117ded | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,y,z>0$. Prove that
\[
\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{z+x}\ge
\frac{x}{x^{2}+yz}+\frac{y}{y^{2}+xz}+\frac{z}{z^{2}+xy}\, .
\] | {
"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 | 09a43ec3-fcce-5e24-9332-7f7e304fca97 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,y,z\in\mathbb R$.
a) Prove that
\[
\bigl(x^2+y^2+z^2-1\bigr)^2+3\ge\bigl(xy+yz+zx-1\bigr)^2.
\]
b) Prove that
\[
\bigl(x^4+y^4+z^4-1\bigr)^2+\bigl(x^3y+y^3z+z^3x+1\bigr)^2+1
\;\ge\;2\,(xy+yz+zx)\,\bigl(x^3y^3+y^3z^3+z^3x^3\bigr)
\]
and determine all triples $(x,y,z)$ for which equality 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 | 57a2e1d2-3f46-52ed-b3de-b7b164f47977 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\Omega_1$ and $\Omega_2$ be two circles intersecting at $X$ and $Y$, with centers $O_1$ and $O_2$, respectively. The line $O_1Y$ meets $\Omega_2$ again at $A$, and the line $O_2X$ meets $\Omega_1$ again at $B$. The line $XA$ meets $\Omega_1$ again at $C$, and the line $YB$ meets $\Omega_2$ again at $D$. Let $(DXO_... | {
"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 | e8350dfd-cec4-5eba-88a5-4fe6454ef55b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | P is a point inside triangle $ABC$ such that $\angle PAC + \angle PBC = \angle PAB + \angle PCB$. Find the locus of $P$. | {
"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 | 9089a456-7a66-5db9-a739-b4935d9502ed | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | A finite set $A$ of positive integers is called **good** if for every element $x\in A$, $x$ divides the sum of all elements of $A$. Prove that for any finite set of positive integers there exists a finite good set $B$ such that $A\subseteq B$. | {
"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 | fe914267-352a-5305-bcfb-ce81aaaaa818 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $O$ be a point inside $\triangle ABC$ such that $OA=OB+OC$. Let $B'$ and $C'$ be the midpoints of the arcs $\widehat{AOC}$ and $\widehat{AOB}$ respectively (midpoints on the circumcircles of $\triangle AOC$ and $\triangle AOB$). Prove that the circumcircles $(C O B')$ and $(B O C')$ are tangent to each other. | {
"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 | 2f9d7945-78c8-5ee9-8e14-79ef04d0e7a6 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ and polynomials $P(x),\,Q(x),\,R(x)$ with positive real coefficients such that $Q(-1)=-1$ and for all positive real numbers $x,y$
\[
f\!\Big(\frac{x}{y}+R(y)\Big)=\frac{f(x)}{Q(y)}+P(y).
\] | {
"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 | b1a9600d-82b9-52b4-8cb9-fc9edb881b4b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\in\mathbb{R}$. Prove that
\[
a(a+b)^{5}+b(b+c)^{5}+c(c+a)^{5}\ge 0\,.
\] | {
"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 | a2ad912d-2359-580c-8e0f-2f7d31f4dcc1 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For fixed positive integers $s$ and $t$, define a sequence $(a_n)_{n\ge1}$ by
\[
a_1=s,\qquad a_2=t,\qquad a_{n+2}= \Big\lfloor\sqrt{\,a_n+(n+2)a_{n+1}+2008\,}\Big\rfloor\qquad\text{for all }n\ge1 .
\]
Prove that the set $\{\,n\in\mathbb N : a_n\neq n\,\}$ is finite; equivalently, show that there exists $N$ (depend... | {
"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 | 6bd28ba4-6dab-56e7-9da5-d424fffd4b1c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\ge0$ and define
\[
x=a+b+c,\qquad y=ab+bc+ca,\qquad z=abc.
\]
Assume that
\[
4x^{4}+25xz+4y^{2}\ge17x^{2}y.
\]
Prove that
\[
\sqrt{9a^{2}+16bc}+\sqrt{9b^{2}+16ca}+\sqrt{9c^{2}+16ab}\le5(a+b+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 | 87cb2f99-96b4-5884-ad34-4ffd49a0f800 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,y,z\ge0$. Prove each of the following six inequalities:
\[
\begin{aligned}
&(1)\quad -(x+y+z)xyz+x^4+yz\bigl(y^2+z^2\bigr)\ge0,\\[2mm]
&(2)\quad -2(x+y+z)xyz+\bigl(x^2+yz\bigr)^2+yz\bigl(y^2+z^2\bigr)\ge0,\\[2mm]
&(3)\quad -2(x+y+z)xyz+x^2\bigl(x^2+y^2\bigr)+yz\,(y+z)(z+x)\ge0,\\[2mm]
&(4)\quad -2(x+y+z)xyz+\big... | {
"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 | cf8f1831-88d0-5c60-b505-fa1d3a8ca824 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n\ge 2$ be an integer and let $1<a_1\le a_2\le\dots\le a_n$ be real numbers satisfying
\[
a_1+a_2+\dots +a_n=2n .
\]
Prove that
\[
a_1a_2\cdots a_{n-1}+a_1a_2\cdots a_{n-2}+\dots +a_1a_2+a_1+2\;\le\;a_1a_2\cdots a_n .
\]
(Equality holds precisely when $a_1=a_2=\dots =a_n=2$.) | {
"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 | e5432033-3be1-5a1e-8f91-355796fb6302 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute triangle with incircle $(I)$.
For each vertex $X\in\{A,B,C\}$ define the circle $(K_X)$ as the unique circle that
* passes through $X$,
* has its centre on the line through $X$ perpendicular to the opposite side, and
* is internally tangent to $(I)$.
Denote by $X_1$ the point of tangency $(K_X... | {
"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 | 3925c951-c0bc-54c6-adca-831c6a7ee406 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
$$f\bigl(x+f(y)\bigr)=f(x+xy)+y\,f(1-x)$$
holds 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 | 17a7d99e-e5ce-533a-a237-84b40d69ec69 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Evaluate \[\displaystyle \int_{0}^{\pi}\frac{\sin (n x)}{1+\cos^{2}x}\,dx\] where \(n\) is an integer. | {
"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 | ba9b51d9-3ab1-5b62-8e0b-58892ecf5d88 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(a,b,c>0\) be real numbers such that \(\min\{ab,bc,ca\}\ge 1\) (equivalently \(ab\ge1,\;bc\ge1,\;ca\ge1\)). Prove that
\[
\sqrt[3]{(a^{2}+1)(b^{2}+1)(c^{2}+1)}\;\le\;\Bigl(\frac{a+b+c}{3}\Bigr)^{2}+1 .
\]
Equality holds when \(a=b=c\ge1\). | {
"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 | e407e85a-760b-5ea0-8a92-61286ef6fce9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given four points \(A,B,C,D\) on a circle \((O)\).
Let \((O_1)\) be the circle through \(A\) and \(B\), \((O_2)\) the circle through \(B\) and \(C\), \((O_3)\) the circle through \(C\) and \(D\), and \((O_4)\) the circle through \(D\) and \(A\).
Define
- \(A'\) as the second intersection of \((O_1)\) and \((O_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 | c4b6d88a-4dac-53de-8c33-dfccd7307008 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\ge0$ satisfy $a+b+c=\sqrt5$. Prove that
\[
(a^2-b^2)(b^2-c^2)(c^2-a^2)\le\sqrt5\, .
\]
Equality occurs for $\bigl(\tfrac{\sqrt5+1}{2},0,\tfrac{\sqrt5-1}{2}\bigr)$ and its cyclic permutations. | {
"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 | e3fc6598-a1dc-5280-9791-98b1230aeab2 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ satisfy $ab+bc+ca=1$. Prove that
\[
\frac{1+a^{2}b^{2}}{(a+b)^{2}}+\frac{1+b^{2}c^{2}}{(b+c)^{2}}+\frac{1+c^{2}a^{2}}{(c+a)^{2}}\ge\frac52\,,
\]
with equality when $a=b=c=\frac1{\sqrt3}$. | {
"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 | 8b52e862-8fd2-51e1-a28c-16c583384947 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In the plane the points with integer coordinates are the vertices of unit squares. The squares are coloured alternately black and white (as on a chessboard). For any pair of positive integers \(m\) and \(n\), consider a right‑angled triangle whose vertices have integer coordinates and whose legs, of lengths \(m\) 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 | 05e09910-a2e2-5483-be17-2e670f6462de | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ with $abc=1$. Prove that
\[
3\;\le\;\sum_{cyc}\frac{a(3a+1)}{(a+1)^2}\;\le\;a+b+c,
\]
where the sum is taken over $a,b,c$ cyclically. | {
"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 | 900c2984-c287-52c6-957f-bc2a62477c9d | [] | AoPS | Nemotron-Math-Proofs-v3 |
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