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 a triangle with circumcircle $\omega$ (center $O$). Let $\omega_A$ be the $A$‑mixtilinear incircle, i.e. the circle inside $\omega$ that is tangent to $\omega$ at a point $A'$ and tangent to the sides $AB$ and $AC$. Define $B'$ and $C'$ analogously as the tangency points of the $B$‑ and $C$‑mixtilinear inc... | {
"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 | 91762a23-8551-52c7-a9c6-03dd519c929b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find the smallest constant $C>0$ such that for any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct) there exist four distinct indices $i,j,k,l$ with
\[
\left| \frac{a_i}{a_j} - \frac{a_k}{a_l} \right| \le C .
\] | {
"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 | b7e142a4-07dc-586a-bc60-565670a9a697 | [] | 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 $\widehat{BAC}$ of $(O)$.
Let $U\neq C$ be the second intersection of the line through $C$ parallel to $BA^+$ with $(O)$.
Let $V\neq B$ be the second intersection of the $B$‑altitude (the line through $B$ perpendicular to $AC$) ... | {
"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 | 62ae8ed3-9658-5005-a47a-aa6d8012ba7c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ satisfying
\[
f\bigl(x+y^{2}f(x^{2})\bigr)=f(xy)^{2}+f(x)
\]
for every $x,y\in\mathbb{R}^+$. | {
"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 | 618632b1-d85e-591d-a48a-e81f39e64c9a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For all positive real numbers \(a,b,c\) prove that
\[
\frac{1}{b(a+b)}+\frac{1}{c(b+c)}+\frac{1}{a(c+a)}\ge \frac{27}{2\,(a+b+c)^{2}}.
\] | {
"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 | a1e04835-fda6-5845-9306-ae8c82c0eee0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(ABCDEF\) be a convex hexagon satisfying the parallelism conditions \(AC\parallel DF,\;BD\parallel AE,\;CE\parallel BF.\) Prove that
\[
AB^{2}+CD^{2}+EF^{2}=BC^{2}+DE^{2}+AF^{2}\,.
\] | {
"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 | 50aab280-812f-5ad4-84e2-e153949f2794 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n\ge 2$ be a positive integer with divisors $1=d_1<d_2<\dots<d_k=n$. Prove that
\[
S:=d_1d_2+d_2d_3+\dots+d_{k-1}d_k
\]
is always less than $n^2$, and determine for which $n$ the sum $S$ is a divisor of $n^2$. | {
"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 | 6cc1bb92-c7c1-59c6-aa3d-4ba2d44db400 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all ordered pairs of prime numbers \((p,q)\) such that both \(p-q\) and \(pq-q\) are perfect squares. | {
"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 | 0cdc8b5a-5271-55bc-9737-903945481537 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a convex quadrilateral. Points $M$ and $N$ lie inside the quadrilateral and points $P$ and $Q$ lie outside it such that the triangles $\triangle MAD$, $\triangle NCB$, $\triangle PAB$, and $\triangle QCD$ are all similar to each other. Prove that quadrilateral $MNPQ$ is a parallelogram. | {
"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 | 0312d53f-aed3-57ca-90da-6d73b75f0208 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\ge0$ with $a+b>0,\;b+c>0,\;c+a>0$. Prove
\[
\sqrt{1+\frac{48a}{\,b+c\,}}+\sqrt{1+\frac{48b}{\,c+a\,}}+\sqrt{1+\frac{48c}{\,a+b\,}}\ge15,
\]
and 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 | 54c1ff43-f866-5cf9-9bea-04a65e8afd59 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$. Prove that
\[
\frac{a^{3}+b^{3}+c^{3}}{3abc}+\frac{a(b+c)}{b^{2}+c^{2}}+\frac{b(c+a)}{c^{2}+a^{2}}+\frac{c(a+b)}{a^{2}+b^{2}}\ge 4,
\]
with equality if and only if $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 | db734708-52bc-5d98-ada4-030ee0f3ae2b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $S$ be a finite set of points in the plane such that for each two distinct points $A$ and $B$ in $S$, the segment $AB$ is a side of a regular polygon whose all vertices are contained in $S$. Find all possible values for the number of elements 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 | 1ea77c8b-edbf-5c39-b64a-c4ff6bd41662 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a non‑isosceles triangle with incircle $(I)$. Denote by $D\in BC$, $E\in CA$, $F\in AB$ the points of tangency of $(I)$ with the sides $BC$, $CA$, $AB$, respectively.
Let $P=DE\cap AB$. Let a line through $C$ intersect $AB$ at $N$ and $FE$ at $M$. Let $Q=PM\cap AC$.
Prove that the line $IN$ is per... | {
"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 | 9fb0fd35-51d2-5a6c-bd5b-1c2e46860234 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find the largest constant \(k\) such that
\[
\sum_{\text{cyc}} a\,(a-b)\,(a-c)\ \ge\ k\sum_{\text{cyc}} \frac{ab}{a+b}\,(a-b)^{2}
\]
holds for all positive real numbers \(a,b,c>0\). | {
"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 | 0dda3e6b-370f-52d2-9969-09888b38b253 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n$ be a positive integer. Prove that
\[
\sum_{k=1}^{n} (-1)^{\big\lfloor k(\sqrt{2}-1)\big\rfloor}\ge 0.
\]
(Here $\lfloor x\rfloor$ denotes the greatest integer $\le x$.) | {
"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 | f965b8e9-6c56-5516-bb36-1f6e7ed517e7 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be positive real numbers such that $a^{2}+b^{2}+c^{2}=3$. Prove that
\[
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+3\;\ge\;2\Bigl(\frac{a+b}{a^{2}+b^{2}}+\frac{b+c}{b^{2}+c^{2}}+\frac{c+a}{c^{2}+a^{2}}\Bigr).\] | {
"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 | 97374bec-78a1-5e75-a8bf-eed49cba0272 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\ge0$. Prove that
\[
\frac{a^{2}}{\,b^{2}-bc+c^{2}\,}+\frac{b^{2}}{\,a^{2}-ac+c^{2}\,}+\frac{c^{2}}{\,a^{2}-ab+b^{2}\,}\ge
2+\frac{abc\,(a^{3}+b^{3}+c^{3})}{3\,(b^{2}-bc+c^{2})(a^{2}-ac+c^{2})(a^{2}-ab+b^{2})}\,.
\] | {
"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 | 305d0de5-f440-5377-9df7-0798bad86703 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine all 4‑tuples \((x_0,x_1,x_2,x_3)\) of pairwise distinct integers such that each \(x_k\) is coprime to \(x_{k+1}\) (indices taken modulo 4) and the cyclic sum
\[
\frac{x_0}{x_1}+\frac{x_1}{x_2}+\frac{x_2}{x_3}+\frac{x_3}{x_0}
\]
is an integer.
Show that there are infinitely many 5‑tuples \((x_0,x_1,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 | 4c616617-c57e-592d-aabf-5ad330505e58 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\triangle ABC$ be an acute triangle with incenter $I$ and circumcenter $O$. The incircle touches $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. Let $A'$ be the reflection of $A$ across $O$ (the antipode of $A$ on the circumcircle). Let $G$ be the second intersection of the circumcircles $(ABC)$ and $(A'E... | {
"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 | e28836c2-4e57-5f63-b6f1-857abd6a8f62 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $P$ be any point in the plane of triangle $ABC$. The lines $AP$, $BP$, $CP$ intersect the circumcircle $(ABC)$ again at $A_1$, $B_1$, $C_1$, and intersect the circumcircles $(BPC)$, $(CPA)$, $(APC)$ again at $A_2$, $B_2$, $C_2$, respectively. Define $A_3$, $B_3$, $C_3$ as the reflections of $A_2$, $B_2$, $C_2$ acro... | {
"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 | ce70119e-59ad-5ecb-82ac-7dee55b6f843 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a parallelogram. Let $(ABC)$ be its circumcircle. Choose a point $P$ on the arc $BC$ of $(ABC)$ that does **not** contain $A$, and choose a point $Q$ on the segment $AC$ such that
\[
\angle PBC = \angle CDQ.
\]
Prove that the circumcircle of $\triangle APQ$, denoted $(APQ)$, is tangent to the line $AB$ 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 | be1b6fe2-667c-55ea-a5f4-65b22230a999 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be positive real numbers with $a+b+c=1$. Prove that
\[
(ab+bc+ca)\Bigl(\frac{a}{b^2+b}+\frac{b}{c^2+c}+\frac{c}{a^2+a}\Bigr)\ge\frac34\,.
\] | {
"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 | b6d2e319-faec-5d94-88b9-0e1e5098a485 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x_1,x_2,\dots,x_n>0$ with $n\ge2$. Prove that
\[
\biggl(\sum_{i=1}^n x_i\biggr)\!\biggl(\sum_{i=1}^n\frac1{x_i}\biggr)-n^2
\;\ge\;(n-1)^2\Biggl[\sum_{i=1}^n\frac{x_i}{\displaystyle\sum_{j\neq i}x_j}
-\frac{n}{\,n-1\,}\Biggr]\;\ge\;0\,.
\]
(Here $\displaystyle\sum_{i=1}^n\frac{x_i}{\sum_{j\neq i}x_j}$ is the cy... | {
"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 | f4218241-9c76-5855-8531-d9ce8cab3cd4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with circumcenter $O$. Let $\Gamma$ be a circle passing through $A$ and intersecting $AB$ again at $E$ and $AC$ again at $F$. Let $P\neq A$ be the second intersection of $\Gamma$ with the circumcircle $(ABC)$. Prove that the reflection of $P$ across the line $EF$ lies on the line $BC$ if and onl... | {
"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 | cad953ef-00dc-5613-bcca-b2a1ecaed865 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with orthocenter $H$. Denote by $D$, $E$, $F$ the feet of the altitudes from $A$, $B$, $C$ respectively (so $D=AH\cap BC$, $E=BH\cap CA$, $F=CH\cap AB$). Let $X=BH\cap DF$ and $Y=CH\cap DE$. Let the line $FE$ intersect the circumcircle of $ABC$ again at the two points $P$ and $Q$. Prove that 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 | 73a8f2dd-1182-57c8-901d-9aa033a2af74 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given $\triangle ABC$ with $AB<AC$, let $(O)$ be its circumcircle. Let $AD$, $BE$, $CF$ be the altitudes, meeting at the orthocenter $H$, where $D\in BC$, $E\in AC$, $F\in AB$.
(a) Let $M,N=EF\cap (O)$, with $M$ on the minor arc $AB$ of $(O)$. Let $I$ be the midpoint of $BC$, and let line $MI$ meet $(O)$ again at $K\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 | a9bc601a-9e06-52b8-a629-eac3274d337b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $f(x)\in\mathbb R[x]$ be a monic polynomial of degree $n$, i.e.
$$f(x)=x^n+a_{n-1}x^{\,n-1}+a_{n-2}x^{\,n-2}+\dots+a_1x+a_0.$$
Prove that there exists an integer $i$ with $1\le i\le n$ such that
$$|f(i)|\ge\frac{n!}{\binom{n}{i}}\,. $$ | {
"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 | 8db96320-b23e-59e1-9a68-942daaa36665 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle. Let $H$, $I$, $O$ denote respectively the orthocenter, the incenter, and the circumcenter of $ABC$. Let $D$, $E$, $F$ be the points where the incircle of $ABC$ touches $BC$, $CA$, and $AB$ (the vertices of the intouch triangle of $ABC$). Define $U=AO\cap CI$ and $M=OI\cap AC$. Prove that the th... | {
"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 | 624d596e-71d2-51f3-b62b-c6a16494ba0e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,\;y,\;z\ge 0$. Prove that
\[
1+\sqrt[3]{(1+x^{3})(1+y^{3})(1+z^{3})}\;\ge\;x+y+z\,.
\] | {
"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 | 9f5769b2-2836-5856-8cab-60d2a6e6b865 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ such that $ab+bc+ca=3$. Prove that
\[
\frac{a+3}{b^2+1}+\frac{b+3}{c^2+1}+\frac{c+3}{a^2+1}\ge 6.
\] | {
"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 | 5c3eed31-8ea8-55a3-9c30-d217e475edec | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,y,z$ be positive real numbers such that $xyz\ge 1$. Prove that
\[
\frac{x^5-x^2}{\,x^5+y^2+z^2\,}+\frac{y^5-y^2}{\,x^2+y^5+z^2\,}+\frac{z^5-z^2}{\,x^2+y^2+z^5\,}\ge 0.
\] | {
"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 | 02238f43-7f2d-5df3-b7b4-adb506e098e8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be integers satisfying $a^2+b^2+c^2-2abc=1$. Prove that at least one of the three numbers $\displaystyle\frac{a+1}{2},\;\frac{b+1}{2},\;\frac{c+1}{2}$ is a perfect square (i.e. equals $k^2$ for some integer $k$). | {
"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 | 2b20fba9-70ea-5531-8b3d-72ae52fd5900 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a tangential quadrilateral with incircle $(I)$. Define $E=AB\cap CD$ and $F=AD\cap BC$. Let $K$ be the Miquel point of the complete quadrilateral formed by the six lines $AB,BC,CD,DA,EF$. Prove that the line $KI$ bisects $\angle BKD$, i.e. $\angle BKI=\angle IKD\,. | {
"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 | d083b9aa-52ac-504c-bb26-118a4f78e219 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | The circles $C_1$ and $C_2$ intersect at distinct points $A$ and $K$. The common external tangent to $C_1$ and $C_2$ that lies nearer to $K$ touches $C_1$ at $B$ and $C_2$ at $C$. Let $P$ be the foot of the perpendicular from $B$ to $AC$, and let $Q$ be the foot of the perpendicular from $C$ to $AB$. Let $E$ and $F$ be... | {
"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 | b7333f7f-5586-5d1d-879a-2642abdb1386 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ satisfy $a+b+c=3$. Prove that
\[
\frac{a+1}{b}+\frac{b+1}{c}+\frac{c+1}{a}\ge 2\,(a^2+b^2+c^2)\,.
\] | {
"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 | 886956b2-6656-5e79-9f21-096819d76104 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,y,z\in\mathbb R^+$ satisfy $x+y+z=xyz$. Prove that
\[
\sqrt{x^2+y^2}+\sqrt{y^2+z^2}+\sqrt{z^2+x^2}+3\sqrt6\le\sqrt8\,xyz.
\] | {
"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 | de9d927a-67ab-5d05-b65a-f7303601b9c8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a fixed convex quadrilateral with $BC=DA$ and $BC$ not parallel to $DA$. Let $E$ be a variable point on $BC$ and $F$ a variable point on $DA$ such that $BE=DF$. Define
\[
P=AC\cap BD,\qquad Q=BD\cap EF,\qquad R=EF\cap AC .
\]
Prove that all circumcircles of $\triangle PQR$ (as $E$ and $F$ vary) share 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 | 70e91ab6-b0f8-5682-bfe0-ee08ac448cc2 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Two circles $\Gamma_1$ and $\Gamma_2$ intersect at $A$ and $B$. A line through $B$ meets $\Gamma_1$ again at $C$ and $\Gamma_2$ again at $D$. Another line through $B$ meets $\Gamma_1$ again at $E$ and $\Gamma_2$ again at $F$. The line $CF$ meets $\Gamma_1$ again at $P$ and $\Gamma_2$ again at $Q$. Let $M$ be the midpoi... | {
"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 | e53aedc2-b078-561c-b24e-0b090b1ee212 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\alpha\in\mathbb C\setminus\{0\}$ and let $A\in\mathcal M_n(\mathbb C)$, $A\neq0$, satisfy
$$A^{2}+(A^{*})^{2}=\alpha\,AA^{*},$$
where $A^{*}=\overline{A}^{\,T}$ is the conjugate transpose of $A$. Prove that $\alpha$ is real, $|\alpha|\le2$, and that $A$ commutes with its conjugate transpose, i.e. $AA^{*}=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 | bf2c22c6-559e-50c6-b8f6-3529772c6148 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In triangle $ABC$ let $O$ be its circumcenter. Let $H$ be the foot of the perpendicular from $A$ to $BC$. Let $T$ be the foot of the perpendicular from $H$ to $AB$, and let $T'$ be the foot of the perpendicular from $H$ to $AC$. If $2\,OT = AC$, prove that $2\,OT' = AB$. | {
"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 | 8ffd735d-c3bc-5304-94ef-2b45d8c69298 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $p=4k+1$ be a prime number. Prove that there exist infinitely many natural numbers $n$ such that $\displaystyle\big\lfloor n\sqrt{p}\big\rfloor$ 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 | 4deec1eb-1c64-54b2-a816-4cbd0339b1db | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that an integer $a$ is a perfect square in $\mathbb Z$ if and only if for every prime $p$ the residue class $\overline a$ of $a$ modulo $p$ is a square (quadratic residue) in the finite field $\mathbb Z/p\mathbb Z$. | {
"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 | 9931851e-77ad-5f8a-960f-c400c02959ea | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Show that there is no integer $n\neq0$ such that the three numbers $2n^{2}+1$, $3n^{2}+1$, and $6n^{2}+1$ are all perfect squares. | {
"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 | 00ce5740-0a39-5f85-8513-cdd2b3974366 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Sides $AB$ and $DC$ of a cyclic quadrilateral $ABCD$ intersect at $Q$, and sides $DA$ and $CB$ intersect at $P$. Let $L$ and $M$ be the midpoints of the diagonals $AC$ and $BD$, respectively, and let $K$ be the orthocenter of triangle $MPQ$. Prove that the four points $P$, $Q$, $K$, and $L$ are concyclic. | {
"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 | bf535150-adfd-5a0d-96f6-8be248c4629e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that there exists a positive integer $N$ such that for every integer $n>N$ there are integers $a_1,a_2,\dots,a_{2018}$ satisfying
\[
n=a_1+a_2+\cdots+a_{2018},\qquad 2018<a_1<a_2<\dots<a_{2018},
\qquad \gcd(a_i,a_j)=1\ \text{for all}\ 1\le i<j\le2018.
\] | {
"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 | 5252031d-b4bf-5cc4-b3fe-9ee35f837eca | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given a tournament with $n$ teams in which each pair of distinct teams plays exactly once and there are no draws, assume that for any two distinct teams there exists a third team that defeated both of them. Determine all integers $n$ for which such a tournament can exist. | {
"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 | d2818978-923c-5640-b55c-33789a87827a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a convex cyclic quadrilateral. The reflections of the diagonal $BD$ across the sides $AB$ and $AD$ intersect at a point that lies on the line $AC$. Given $AB=4$, $BC=3$, and $CD=2$, find the length $AD$. | {
"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 | 4116e68d-ecb2-5317-b7b7-732760e4fce9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given a positive real number $r$ (the inradius) and a positive real number $p$ (the perimeter) for which a triangle exists, construct with straightedge and compass a triangle $ABC$ whose inradius is $r$ and whose perimeter is $p$. | {
"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 | 5b4b3fa0-95a0-53a0-91e0-f823a31cc097 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Suppose $a_1,\dots,a_n$ are rational numbers such that for every positive integer $m$ the sum $a_1^m+\cdots+a_n^m$ is an integer. Prove that each $a_i$ 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 | 0565fc8f-4f00-58f0-a37b-4e28d04af7d9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCDE$ be a convex pentagon such that $BC=DE$. Assume there is a point $T$ inside $ABCDE$ with $TB=TD$, $TC=TE$ and $\angle ABT=\angle TEA$. Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$, respectively, and assume the points appear on line $AB$ in the order $P,B,A,Q$. Let line $AE$ intersect $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 | c045cf6b-a264-513a-a3b3-6aef6974cc9e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ satisfy
\[
2a^3b+2b^3c+2c^3a=a^2b^2+b^2c^2+c^2a^2.
\]
Prove that
\[
2ab(a-b)^2+2bc(b-c)^2+2ca(c-a)^2\ge (ab+bc+ca)^2.
\] | {
"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 | 4e6897dc-4f60-5354-83b7-c2772b07ca9d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In an acute triangle $BDE$, let $\omega$ be its circumcircle. Let $C$ be the intersection of the two tangents to $\omega$ at $D$ and $E$. Choose points $G$ on the extension of $BD$ beyond $D$ and $F$ on the extension of $BE$ beyond $E$, both lying outside $\omega$, such that the line $GF$ passes through $C$. 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 | 926f9894-31ab-5202-a285-cbcac81d2d39 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $I$ and $O$ be the incenter and circumcenter of $\triangle ABC$, respectively. Let $K=IO\cap BC$. Let $\omega$ be the circle tangent to $AB$, $AC$, and the circumcircle $(O)$; denote by $T\neq A$ the second intersection of $\omega$ with $(O)$. Let $\ell$ be the line tangent to the circumcircle of $\triangle ITK$ at... | {
"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 | 28a97821-5002-5a9a-8979-3196e2bc751d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with incenter $I$.
For each of the three triangles $IBC$, $ICA$, and $IAB$, let the incircle of that triangle touch the two sides that contain $I$ at points
\[
I_a,\;I_a' \text{ on } IB,IC \text{ respectively (for } IBC),\\
I_b,\;I_b' \text{ on } IC,IA \text{ respectively (for } ICA),\\
I_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 | ea17d08e-8ac0-588c-bf14-a77fba11e2b2 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | On a circular table sit $2n$ people equally spaced. $m$ cookies are placed arbitrarily among them. A move consists of a person eating one of his own cookies and then giving a (different) cookie to an adjacent neighbor. Choose an arbitrary person $A$. Determine the smallest integer $m$ such that, for every possible init... | {
"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 | b8fb3cb7-b1e3-5848-a689-4be36575c14a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(a,b,c>0\) be real numbers such that \(a^{3}+b^{3}+c^{3}=3\). Prove that
\[
\frac{a^{4}}{a+b^{6}}+\frac{b^{4}}{b+c^{6}}+\frac{c^{4}}{c+a^{6}}\ge \frac{3}{2}\,.
\] | {
"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 | 1eeade2d-f360-5914-9395-6f8470cc3591 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle. Points $M\in AB$ and $N\in AC$ satisfy $MN\parallel BC$. Choose points $K$ and $L$ on the line $MN$ such that the lines $BL$ and $CK$ intersect at $P$. Let $\omega_1$ be the circumcircle of $\triangle MPK$ and $\omega_2$ be the circumcircle of $\triangle NPL$. Let $Q\neq P$ be the second common... | {
"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 | b4fa315a-de65-5827-9ce5-16a60338d642 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute‑angled triangle inscribed in the circle $c(O,R)$ with $AB<AC<BC$, and let $c_1$ be the incircle of $ABC$ which meets $AB$, $AC$, $BC$ at $F$, $E$, $D$, respectively. Let $A'$, $B'$, $C'$ be points on $c$ such that the quadrilaterals $AEFA'$, $BDFB'$, $CDEC'$ are cyclic.
1) Prove that $DEA'B'$ is... | {
"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 | 5abc170c-a906-579d-985a-09cb23c74f4e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Call a subset $X\subseteq\mathbb{R}^2$ a “Y‑set’’ if it is the union of three straight line segments that share a common endpoint (the three segments may meet at any angles and each may have any finite length). Prove that any collection of pairwise disjoint Y‑sets is countable. | {
"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 | 09ea4a2f-8607-5cb5-95f8-7a970cf9385d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given positive real numbers $a,b,c$, prove that
\[
\frac{a+b}{b+c}\cdot\frac{a}{2a+b+c}+\frac{b+c}{c+a}\cdot\frac{b}{2b+c+a}+\frac{c+a}{a+b}\cdot\frac{c}{2c+a+b}\ge\frac34,
\]
and determine the equality case. | {
"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 | 9ac3939d-fe55-551a-a8f8-f1b260791566 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $N$ be the Nagel point of triangle $ABC$. Let $D=BN\cap AC$ and $E=CN\cap AB$. Let $M$ and $T$ be the midpoints of $BE$ and $CD$, respectively. Let $P$ be the second intersection point of the circumcircles of triangles $BEN$ and $CDN$ (the first intersection is $N$). Let $l_1$ be the line through $M$ perpendicular ... | {
"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 | 5634d5dc-b304-5df0-b84a-5103bdd9dacd | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
$$f\bigl(x^2+2f(xy)\bigr)=x\,f(x+2y)\qquad\forall\,x,y\in\mathbb{R}.$$ | {
"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 | acf7382f-ee43-5a2d-b789-c02da4d049d9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is called good if $\operatorname{lcm}(A_1)=\gcd(A_2)$. Determine the smallest integer $n$ for which there exists a set of $n$ positive integers that has exactly $2015$ good partitions. | {
"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 | 04500a18-acdb-5ebe-8976-d2fefe4a7dee | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In any triangle \(ABC\) with side lengths \(a,b,c\), medians \(m_a,m_b,m_c\), semiperimeter \(s=\dfrac{a+b+c}{2}\) and inradius \(r\), prove that
\[
\frac{m_a}{a}+\frac{m_b}{b}+\frac{m_c}{c}\;\le\;\frac{s}{2r}.
\]
Equality holds only for an equilateral triangle. | {
"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 | b26ae536-136f-5a37-a2a6-c2150205b9c3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be positive real numbers. Prove that
\[
\left(\sum_{cyc}\frac{a}{\,b+c\,}\right)^2\le\left(\sum_{cyc}\frac{a^2}{\,bc+c^2\,}\right)\left(\sum_{cyc}\frac{a}{\,a+b\,}\right).
\] | {
"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 | 056c17dd-1e22-5096-87a9-2f6c89bcd437 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ and $ABD$ be coplanar triangles sharing side $AB$ and having equal perimeters. Let $P$ be the intersection of the internal angle bisectors of $\angle CAD$ and $\angle CBD$. Prove that $\displaystyle \angle APC = \angle BPD\,. $ | {
"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 | 56d2d31f-fc1b-5cfe-b70c-1899c0a20c5b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $p,q>1$ be coprime positive integers. Call a positive integer $a$ **bad** if it cannot be expressed as $a=px+qy$ with $x,y\in\mathbb Z_{\ge0}$. Define
\[
S(p,q)=\sum_{\substack{a>0\\ a\text{ bad}}} a^{2019}.
\]
Prove that there exists a positive integer $n$, independent of $p$ and $q$, such that for every pair ... | {
"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 | c7fbdab8-08db-53d7-8ab2-bf64493e1dc7 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute triangle with circumcenter $O$, orthocenter $H$, and circumcircle $\Gamma$. Let $N$ be the midpoint of $OH$.
The tangents to $\Gamma$ at $B$ and $C$ intersect at a point $D_{A}$; let $L_{A}$ be the line through $H$ perpendicular to $AN$. The three lines $D_{A}B$, $D_{A}C$, and $L_{A}$ form a tri... | {
"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 | e92a986c-2e04-576d-b174-42e2980a4e40 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that
$$f(x)f(y)+f(x+y)=f(xy)+f(x)+f(y)$$
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 | 40da0102-98bc-5fd1-9852-f9a4bee1cca4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,\;y,\;z>0$. Prove that
\[
\frac{x^{2}y^{5}z^{5}}{x^{7}z^{5}+x^{2}y^{10}+y^{5}z^{7}}+\frac{y^{2}z^{5}x^{5}}{y^{7}x^{5}+y^{2}z^{10}+z^{5}x^{7}}+\frac{z^{2}x^{5}y^{5}}{z^{7}y^{5}+z^{2}x^{10}+x^{5}y^{7}}\le 1.
\]
(The sum is taken cyclically over $x,\;y,\;z$.) | {
"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 | 71db133a-49c4-515b-8400-fe20cfcb1be8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,y,z>0$ such that $xyz=1$. Prove that
\[
\frac{1}{(1+x)^{3}}+\frac{1}{(1+y)^{3}}+\frac{1}{(1+z)^{3}}
\ge
\frac{xy(x+y)+yz(y+z)+zx(z+x)}{2\,(x+y)(y+z)(z+x)}\,.
\] | {
"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 | 4b4e6fa2-7fa6-584c-9d46-3f5655cb7878 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For which real numbers $k>0$ is it possible to dissect a $1\times k$ rectangle into two similar, but noncongruent, polygons? | {
"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 | 1eaf67b5-7328-5adc-b374-9f8da71639b1 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,y,z\ge 0$. Prove that
\[
\sqrt{y^{2}+z^{2}+4(x-y)(x-z)}+\sqrt{z^{2}+x^{2}+4(y-z)(y-x)}+\sqrt{x^{2}+y^{2}+4(z-x)(z-y)}\ge \sqrt{2}\,(x+y+z).
\]
Equality holds if and only if $x=y=z\,$. | {
"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 | b2c2bdce-a1b6-54ff-bc6e-3e1cd180a0e0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that
\[
\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\;\ge\;\frac{1}{\,ab+bc+ca-2\sum_{cyc}(ab-bc)^2\,}.
\] | {
"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 | d4b78fca-aa8e-5093-863f-c19715d1254e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ be the side lengths of a triangle (so $a+b>c$, $a+c>b$, $b+c>a$). Let $r_a,r_b,r_c$ denote the exradii opposite $A,B,C$ respectively, where $r_a=\dfrac{\Delta}{s-a}$ with $s=\frac{a+b+c}{2}$ and $\Delta$ the area of the triangle. Let $m_a,m_b,m_c$ denote the lengths of the medians from $A,B,C$, where $m_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 | b26b6df0-af8d-5643-adf4-020b4a523a07 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(a,b,c\) be non‑negative real numbers, not two of which are zero. Prove that
\[
\frac{1}{(a+2b)^{2}}+\frac{1}{(b+2c)^{2}}+\frac{1}{(c+2a)^{2}}\ge\frac{1}{ab+bc+ca}\,.
\] | {
"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 | 96d80864-5ec4-55a0-9a56-668ce1205135 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be real numbers such that $1\le a,b,c\le 2$ and $a^{2}+b^{2}+c^{2}=6$. Find the maximum value of
$$P=\Bigl(\frac{4bc^{2}}{a+b}+\frac{4ca^{2}}{b+c}+\frac{4ab^{2}}{c+a}\Bigr)+ab+bc+ca.$$ | {
"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 | 993320b5-8d80-539f-8739-2c668b31b5f0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $A$ and $B$ be two fixed points and let $C$ be a variable point such that $\angle ACB = \alpha$, where $\alpha$ is a constant in $[0^\circ,180^\circ]$. The incircle of $\triangle ABC$ has incenter $I$ and touches $AB$, $BC$, $CA$ at $D$, $E$, $F$, respectively. Lines $AI$ and $BI$ intersect $EF$ at $M$ and $N$, res... | {
"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 | c4f56495-e958-5b94-b0fc-4f7dd4bd17f3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ such that $abc=1$. Prove that
$$\sqrt[4]{6a^2+3bc}+\sqrt[4]{6b^2+3ca}+\sqrt[4]{6c^2+3ab}\le (ab+bc+ca)\sqrt{\sqrt a+\sqrt b+\sqrt c}\,. $$ | {
"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 | 1525a62e-dddb-564b-b6b1-3477c06fd81b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In a scalene triangle $ABC$, assume the side $BC$ is the shortest. Let points $M$ on $AB$ and $N$ on $AC$ satisfy $BM=CN=BC$. Denote by $I$ and $O$ the incenter and circumcenter of $\triangle ABC$, and by $D$ and $E$ the incenter and circumcenter of $\triangle AMN$. Prove that the lines $IO$ and $DE$ intersect at a poi... | {
"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 | 473fd268-3c15-53b6-94da-0c9d34ce56a1 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with $AB<AC$. Let $BX$ and $CY$ be the altitudes from $B$ and $C$, respectively, with $X\in AC$ and $Y\in AB$, and let $H=BX\cap CY$ be the orthocenter. Denote by $M$ and $N$ the midpoints of $AB$ and $AC$. Let $R=BX\cap MN$ and $S=CY\cap MN$. Let $\omega$ be the circumcircle of $\triangle AXY$.... | {
"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 | f4331ae0-535b-5831-ab54-94da9bcb965a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $d(n)$ denote the number of positive divisors of a positive integer $n$ (including $1$ and $n$). Find all positive integers $n$ such that
\[n+d(n)=d(n)^2.\] | {
"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 | 8ca4bad3-ceff-5f38-86d0-47768c731a14 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $f:\mathbb{R}\to\mathbb{R}$ satisfy
$$f\!\left(\frac{x+y}{2}\right)\le\frac{f(x)+f(y)}{2}\qquad\forall\,x,y\in\mathbb{R},$$
(i.e. $f$ is midpoint‑convex) and assume there exists a function $F$ on $\mathbb{R}$ such that $F'(x)=f(x)$ for all $x$. Prove that $f$ is continuous on $\mathbb{R}\,.$ | {
"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 | a28fc86c-5b20-55ef-b53e-7529a2fdac67 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that the sequence
\[
a_n=(n!-1)n!+1 = (n!)^2-n!+1\qquad (n\in\mathbb{Z}_{>0})
\]
contains infinitely many composite numbers; i.e., there are infinitely many positive integers \(n\) for which \(a_n\) is not prime. | {
"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 | ad4eeb81-0583-56db-a6d7-d13d415cf0c8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For positive real numbers $a,b,c$, prove that
\[
\sqrt{20(a^{2}+18ab+b^{2})}+\sqrt{20(b^{2}+18bc+c^{2})}+\sqrt{20(c^{2}+18ca+a^{2})}+7\sqrt{3(a^{2}+b^{2}+c^{2})}\le 27(a+b+c)\,.
\] | {
"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 | fd4ec7f4-437c-556f-aaef-4a6712c91fd4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ and define
\[
A=\sum_{\text{cyc}}\frac{bc}{(b+c)^{2}},\qquad
B=\sum_{\text{cyc}}\frac{bc}{(b+a)(c+a)}.
\]
Prove the three inequalities
\[
\begin{aligned}
1)&\;A\le\frac34\le B,\\[2mm]
2)&\;4A+8B\le 9,\\[2mm]
3)&\;A+3B\ge 3.
\end{aligned}
\] | {
"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 | 59d89336-7af6-558f-ac3e-24894707f2ac | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For $x_1,x_2,\dots,x_n>0$ with $x_1+x_2+\cdots+x_n=1$, find the minimum value of
\[
\Bigl(\frac1{x_1}+\frac1{x_2}+\dots+\frac1{x_n}\Bigr)\,(1-x_1)(1-x_2)\dots(1-x_n)\,.
\] | {
"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 | 12cb8d8f-748e-543e-960b-dca4e56763ee | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In an acute triangle \(ABC\), let \(D\), \(E\), \(F\) be the feet of the altitudes from \(A\), \(B\), \(C\) onto \(BC\), \(CA\), \(AB\) respectively. Let the line through \(D\) parallel to \(EF\) intersect \(AC\) at \(Q\) and \(AB\) at \(R\). Let \(P=EF\cap BC\). Prove that the circumcircle of triangle \(PQR\) passes 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 | e01c0043-6373-5659-99c2-34c57ed9fe9c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given an integer $n\ge 3$, color each unit square of an infinite square grid either black or white. A $3\times3$ block of unit squares is called *isolated* if its central unit square has the opposite color of each of the eight surrounding squares (i.e., the center is white and the other eight are black, or the center 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 | 97bd687e-fd9b-544f-bf3c-095ad0661fd3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute, scalene triangle with circumcircle $\Omega$.
Let $A^{+},B^{+},C^{+}$ be the midpoints of the arcs $BAC$, $CBA$, $ACB$ of $\Omega$ (the arcs containing the opposite vertex).
Let $A_{2},B_{2},C_{2}$ be the midpoints of the minor arcs $BC$, $CA$, $AB$, respectively.
Define
\[
A_{1}=BC^{+}\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 | 797472aa-204d-5667-8c90-c33eadb08659 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $O$ be the circumcircle of $\triangle ABC$. Let $W_a$ be the $A$‑mixtilinear incircle of $\triangle ABC$, which is tangent to $AB$ at $P$, to $AC$ at $Q$, and to $O$ at $D$. Define $X$ as the intersection of lines $PQ$ and $AD$. Prove that
\[
\angle BXP = \angle CXQ.
\] | {
"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 | e7328a65-5d65-5bf7-94dd-a9b9437bf2a8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $p$ be a prime and $n$ a positive integer. Define $d=\gcd(n,p-1)$. Prove that the set
\[
\{\,a_1^n+a_2^n+\dots +a_d^n \pmod p \mid a_1,a_2,\dots ,a_d\in\mathbb Z/p\mathbb Z\,\}
\]
contains every residue class modulo $p$; i.e. for each $r\in\{0,1,\dots ,p-1\}$ there exist $a_1,\dots ,a_d\in\mathbb Z/p\mathbb Z$ suc... | {
"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 | 9237a745-efb0-557e-8444-790a1382172e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a$, $b$, $c$ be positive real numbers. Prove
\[
(abc)^{2}(a+b+c)^{3}(b+c-a)(c+a-b)(a+b-c)\ge (a^{2}+b^{2}+c^{2})^{3}(b^{2}+c^{2}-a^{2})(c^{2}+a^{2}-b^{2})(a^{2}+b^{2}-c^{2}) .
\]Equality occurs when $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 | 31e0a415-5929-5b9b-929d-0b804f94a82a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be real numbers such that $a+b+c=0$. Prove that
\[
\sqrt{a^{2}+4}+\sqrt{b^{2}+4}+\sqrt{c^{2}+4}
-\bigl(a+\sqrt{a^{2}+4}\bigr)\bigl(b+\sqrt{b^{2}+4}\bigr)\bigl(c+\sqrt{c^{2}+4}\bigr)\ge -2.
\] | {
"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 | 3061566c-cf21-5dc0-93dd-22ca1c564c42 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
\[
f\bigl(f(x)^2+xy\bigr)=x\bigl(f(x)+f(y)\bigr)
\]
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 | 31ba3478-930c-5f01-96dc-9059344f12f7 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all integers $x,y,z\ge0$ such that $x!\,y! = z!$. | {
"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 | 89f5e470-c004-5ab8-9a36-2af42b42bed7 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all odd positive integers $n$ such that for any pair of relatively prime divisors $a$ and $b$ of $n$ (i.e. $a\mid n$, $b\mid n$, $\gcd(a,b)=1$), the number $a+b-1$ also divides $n$. | {
"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 | cc689036-047e-5736-804e-d9917523090f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a_1>0$, $a_2>0$, and for $n\ge 2$ define
\[
a_{n+1}=\frac{2}{a_n+a_{n-1}}.
\]
Show that the sequence $\{a_n\}$ is convergent. | {
"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 | c5a8da59-532f-5753-896a-99543e9a801d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all continuous functions $f:\mathbb{R}\to\mathbb{R}$ such that
\[
f\bigl(x+2f(y)\bigr)=f(x)+f(y)+y\qquad\text{for all real }x,y.
\] | {
"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 | 1cf1431f-3562-5c55-a00f-0692ab34b06b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ satisfy $a^2+b^2+c^2=3$. Prove that
\[
\frac{a^3}{b^2-bc+c^2}+\frac{b^3}{c^2-ca+a^2}+\frac{c^3}{a^2-ab+b^2}\;\ge\;\sqrt{3\sqrt[3]{3\,(ab+bc+ca)^2}}\,.
\]
Equality holds 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 | 8a8e40e2-83fb-5541-82d6-ec503c6f3b25 | [] | AoPS | Nemotron-Math-Proofs-v3 |
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