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 $n>3$ be an integer and let $x_1,x_2,\dots,x_n\ge0$ satisfy
\[
x_1+x_2+\cdots+x_n=2.
\]
Find the minimum value of
\[
S=\frac{x_2}{1+x_1^2}+\frac{x_3}{1+x_2^2}+\cdots+\frac{x_1}{1+x_n^2},
\]
where the indices are taken cyclically ($x_{n+1}=x_1$). | {
"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 | c85b6be8-b274-5821-bd40-b3f6afcb4702 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $A$, $B$, $C$ be three distinct collinear points. Construct a point $D$ on the same line such that the cross‑ratio
\[
(A,B,C,D)=\frac{CA}{CB}\Big/\frac{DA}{DB}=3,
\]
i.e. $\displaystyle\frac{DA}{DB}=\frac1{3}\cdot\frac{CA}{CB}\,. $ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 495551c4-e086-5d28-a2e0-847b1aaea5fb | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\ge0$ with $a+b+c=3$ and let $n\ge2$. Prove that
\[
2\bigl(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}-3\bigr)\;\ge\;3n\left(\sqrt[n]{\frac{5ab+4ac}{9}}+\sqrt[n]{\frac{5bc+4ba}{9}}+\sqrt[n]{\frac{5ca+4cb}{9}}-3\right).
\]
Equality occurs when $a=b=c=1$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 053b4af0-c8c1-5675-a91c-cfab5ddd54f5 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$. Prove that
\[
\frac{(a+b)(b+c)}{c^3+a^3}+\frac{(b+c)(c+a)}{a^3+b^3}+\frac{(c+a)(a+b)}{b^3+c^3}\ge\frac{2\,(a+b+c)^3}{\bigl(a^2+b^2+c^2\bigr)^2}\,.
\] | {
"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 | 029bf7ff-39bf-585e-b5b2-52d06b233a45 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\mathbb{Z}^+$ denote the set of positive integers. Suppose a function $f:\mathbb{Z}^+\to\mathbb{Z}^+$ satisfies
1. For any finite sequence $a_1,a_2,\dots ,a_k$ of positive integers,
\[
f(a_1)+f(a_2)+\dots+f(a_k)\mid f(a_1+a_2+\dots +a_k);
\]
2. There exists a positive integer $m$ such that $f(m)\neq m\,f(1)$... | {
"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 | 70b99507-88ac-5c97-881a-e74a96375dcd | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | A circle passes through vertex $B$ of triangle $ABC$, meets side $AB$ again at $K$ and side $BC$ again at $L$, and is tangent to side $AC$ at its midpoint $M$. Let $N$ be a point on the arc $BL$ of this circle that does **not** contain $K$, chosen so that $\angle LKN=\angle ACB$. Given that triangle $CKN$ is equilatera... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a98954d0-1ad2-5e0f-b4d2-f775e66f2dce | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with $\angle A<60^\circ$. Let $X$ and $Y$ be points on sides $AB$ and $AC$, respectively, such that $CA+AX=CB+BX$ and $BA+AY=BC+CY$. Let $P$ be the point in the plane for which $PX\perp AB$ and $PY\perp AC$. Prove that $\angle BPC<120^\circ$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 9359ac84-41a0-5881-bd11-3dcd8ba52be2 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all positive integers $(p,a,b,c)$ with $p$ a prime that satisfy each of the following equations (treated separately):
1. $p^a+p^b=p^c$
2. $p^a+p^b=c^p$
3. $p^a+b^p=c^p$
4. $a^p+b^p=p^c$
5. $p^a+b^p=p^c$ | {
"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 | 1d7d8560-3901-555a-b802-ad31cd03a39e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine whether there exists a function $f:\mathbb{Z}_{+}\to\mathbb{Z}_{+}$ such that for every positive integers $x,y,z$,
$$f(f(x)) + f(y) + z \mid \bigl(f(f(z))\bigr)^2 + \bigl(f(x)\bigr)^2 + y^2.$$ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 6f37362e-56f9-5dd4-bf8b-74e6992996ff | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | The vertices $A,B,C$ of an acute‑angled triangle $ABC$ lie on the sides $B_1C_1$, $C_1A_1$, $A_1B_1$ respectively of a triangle $A_1B_1C_1$ which is similar to $ABC$ (i.e. $\angle A=\angle A_1$, $\angle B=\angle B_1$, $\angle C=\angle C_1$). Let $H$ and $H_1$ be the orthocenters of $ABC$ and $A_1B_1C_1$, and let $O$ 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 | 41361cca-1e11-5987-9ad3-8a0d37a2819c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In an acute‑angled triangle $ABC$ (i.e., $0<A,B,C<\tfrac{\pi}{2}$ and $A+B+C=\pi$), prove the chain of inequalities
\[
\sum_{\text{cyc}}\frac{\cos A\cos B}{\cos C}\;\ge\;2-\sqrt{2\cos A\cos B\cos C}\;\ge\;2-\frac{r}{R},
\]
where $r$ and $R$ denote respectively the inradius and circumradius of $ABC$. | {
"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 | 3f3de837-eee6-55b7-a33f-af847871ce8d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$, $$f(x^{2}+y+f(y))=2y+f(x)^{2}\,.$$ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 1ab48a69-37e5-504a-9b60-95140c5a3efe | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Consider an isosceles triangle $ABC$ with $AB=AC$. Let $\omega(ABC)$ be its circumcircle. Let $D$ be the intersection of the tangents to $\omega(ABC)$ at $B$ and $C$. Choose a point $F$ on the arc $AB$ of $\omega(ABC)$ that does not contain $C$. Let $K=AF\cap BD$ and let $L=AB\cap CF$. Denote by $T$ and $S$ the centers... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 5addbaea-d3a7-56ca-a912-17ac365e8e98 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Two positive integers $p,q\in\mathbb Z^+$ are given. A blackboard initially contains $n$ positive integers (a multiset). An operation consists of selecting two equal numbers $a,a$ on the board and replacing them by $a+p$ and $a+q$, respectively. Determine the smallest integer $n$ for which there exists an initial co... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 31fbafda-c55d-58c4-9aee-9746208a3e41 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Call a sequence of positive integers \(\{a_n\}_{n\ge1}\) **good** if for any distinct positive integers \(m,n\) one has
\[
\gcd(m,n)\mid a_m^{2}+a_n^{2}\quad\text{and}\quad\gcd(a_m,a_n)\mid m^{2}+n^{2}.
\]
A positive integer \(a\) is called **\(k\)-good** if there exists a good sequence with \(a_k=a\).
Does there... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 7af3a3d4-52a1-5013-9bde-f6f050d7e715 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $m>1$ be a fixed integer. Define a sequence $\{x_i\}_{i\ge0}$ by
\[
x_i=\begin{cases}
2^{\,i}, & 0\le i\le m-1,\\[4pt]
\displaystyle\sum_{j=1}^{m}x_{\,i-j}, & i\ge m.
\end{cases}
\]
Determine the largest integer $k$ for which there exists a block of $k$ consecutive terms of the sequence that are all divisible 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 | 8b5a1249-df68-5742-b463-1ba6ebf14b91 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a fixed non‑degenerate triangle. Through the vertices $A$, $B$, $C$ draw lines $d_a$, $d_b$, $d_c$ respectively such that $d_a\parallel d_b\parallel d_c$ (i.e. the three lines have a common direction). Let $l_a$, $l_b$, $l_c$ be the reflections of $d_a$, $d_b$, $d_c$ in the opposite sides $BC$, $CA$, $AB$ ... | {
"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 | 9a9cf7b1-5015-5332-b509-ede3cd062737 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all ordered pairs $(m,n)$ of positive integers such that $m^{n}+n^{m}=m!+n!$. | {
"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 | b1d9bfc3-094e-5cbc-9d9e-456a4623b794 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $u(x,y)$ satisfying the partial differential equation
\[ u\frac{\partial^2 u}{\partial x\partial y} = \frac{\partial u}{\partial x}\frac{\partial u}{\partial y}. \] | {
"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 | b5f244c5-8cfa-5656-816d-7a4820abcdf0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | There are $2n$ students in a school ($n\in\mathbb N$, $n\ge2$). Each week exactly $n$ distinct students go on a trip (i.e., each trip corresponds to an $n$‑element subset of the $2n$ students). After a number of trips the condition holds that every unordered pair of students has traveled together on at least one trip. ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | b6f8010a-affc-5602-a22d-42df4c2de99e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Consider an $n\times n$ board of unit squares. The main diagonal consists of the $n$ squares from the top‑left corner to the bottom‑right corner. An L‑tromino (three unit squares forming a $2\times2$ square missing one corner) may be rotated. Determine for which integers $n\ge 2$ it is possible to tile every non‑dia... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 1dbb448a-8c82-5105-a835-4adefe5f705f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a non‑degenerate triangle and $P$ an arbitrary point in the plane.
For each vertex $X\in\{A,B,C\}$ do the following:
* Reflect $X$ across the line $PY$ where $Y$ is the other vertex of the triangle, obtaining two points $X_y$ and $X_z$ (for $X=A$ these are $A_b$, the reflection of $A$ in $PB$, and $A_c$,... | {
"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 | 7fb19b5b-7525-5afe-b69b-02c70ce7129a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $P$ be a convex $n$‑gon. A triangulation of $P$ is a division of $P$ into $n-2$ triangles by $n-3$ non‑intersecting diagonals. A flip (move) is the operation that, given two adjacent triangles $ABD$ and $BCD$ sharing the diagonal $BD$, replaces $BD$ by the other diagonal $AC$, producing the triangles $ABC$ and $ACD... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 44ba7f3c-8a4c-5fa9-bba9-63bf25bfef96 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $P$ be an arbitrary point inside $\triangle ABC$. Let $A_1B_1C_1$ be the pedal triangle of $P$ with $A_1\in BC$, $B_1\in CA$, $C_1\in AB$. Let $O$ be the circumcenter of $\triangle A_1B_1C_1$. Define $A_2$, $B_2$, $C_2$ as the midpoints of $PA$, $PB$, $PC$, respectively. Let $A_3=A_1O\cap B_1C_1$, and define $B_3$,... | {
"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 | 6d51a99d-c93f-5650-aa5c-b1abb477f988 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Consider a row of 203 cells numbered from left to right. Initially the leftmost cell (cell 1) contains 203 tokens and all other cells are empty. In one move you may either
1. take a single token from a cell and move it to an adjacent cell (left or right), or
2. take exactly 20 tokens from a cell and move all of 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 | 516bda1f-cbcf-5b45-b326-37b22310cc3c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For all positive real numbers $a,b,c>0$, prove that
$$a^3+b^3+c^3+3abc\;\ge\;ab\sqrt{2(a^2+b^2)}+bc\sqrt{2(b^2+c^2)}+ca\sqrt{2(c^2+a^2)}.$$ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | f6ca6308-701e-5606-8e9c-3639c1cbea50 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that for all positive real numbers $x$ and $y$,
$$f(x+f(y)) + x = f(x+f(x)) + y.$$ | {
"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 | cd7f4634-0da5-513e-a107-ea6163c36d97 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with incenter $I$. Let $\ell$ be the line through $I$ perpendicular to $BC$, and let $\ell$ intersect the circumcircle of $ABC$ again at points $P$ and $Q$. Let $AP$ meet $BC$ at $D$ and $AQ$ meet $BC$ at $E$. Prove that the four points $A,\ I,\ D,\ E$ 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 | c7f36303-5214-5a20-bd56-a10680696d7d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all sequences of positive integers $(a_n)_{n\in\mathbb Z}$ satisfying
$$a_n\,a_{n+3}=a_{n+1}\,a_{n+2}+1$$
for every integer $n$. | {
"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 | bec33df9-17cf-53ff-b074-5d08d16a85d5 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n$ be a positive integer. For $1\le i,j\le n$, let $a_{i,j}>0$ be real numbers satisfying $a_{i,j}\,a_{j,i}=1$ for all $i,j$. Define $c_i=\displaystyle\sum_{k=1}^{n}a_{k,i}$ for each $i$. Prove that
\[
\sum_{i=1}^{n}\frac{1}{c_i}\le 1.
\] | {
"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 | 7c1c27b1-56c7-5a2a-bcf3-e442577429cf | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ satisfy $a^{3}+b^{3}+c^{3}+abc=4$. Prove that
\[
\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}+3abc\ge 6\,.
\] | {
"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 | f420f9a1-5ffb-5554-850c-f11224b56822 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$. Prove that
$$\sqrt{\frac{(a+b)(b+c)(c+a)}{8abc}}\;\ge\;1+\frac35\Bigl(1-\sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}\Bigr).$$ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 6cbd63dd-dd22-5417-80d4-b237572f7dcc | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$. Prove that
\[
\frac{1}{\sqrt{\frac{1}{b^2}+\frac{3}{c^2}}}+\frac{1}{\sqrt{\frac{1}{c^2}+\frac{3}{a^2}}}+\frac{1}{\sqrt{\frac{1}{a^2}+\frac{3}{b^2}}}\le\frac{\sqrt{3(ab+bc+ca)}}{2}\,.
\] | {
"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 | d482f65f-3ffc-52ea-be16-be89a95703d8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
\[
f(1+xy)-f(x+y)=f(x)f(y)\qquad\text{for all }x,y\in\mathbb{R},
\]
and $f(-1)\neq0\,. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | f67f8d4e-e998-5757-bf6b-1b3c09db7c94 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n\ge2$ be an integer and let $x_1,x_2,\dots,x_n$ be positive real numbers such that
\[
\sum_{i=1}^{n}x_i=1,\qquad \sum_{i=1}^{n}\frac{x_i^{2}}{x_{i+1}}=2,
\]
where the indices are taken cyclically ($x_{n+1}=x_1$). Prove that
\[
\sum_{i=1}^{n}\sqrt{x_i^{2}+x_{i+1}^{2}}\le\frac53\, .
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 780c73c8-5376-5b2a-a41d-933878d257cb | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $P$ be a set of $n$ points in $\mathbb{R}^3$, no four of which are coplanar. The points are partitioned into two subsets $\mathcal A$ and $\mathcal B$. An $\mathcal{AB}$‑tree is a collection of $n-1$ segments each joining a point of $\mathcal A$ to a point of $\mathcal B$, such that the segments form a tree (i.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 | c647a262-282a-5a59-95fe-70ee63c512fc | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $P\in\mathbb Z[x]$ be a polynomial of degree at least $4$. An integer $x$ is called $P$‑representable if there exist integers $a,b$ such that $x=P(a)-P(b)$. Suppose that for every integer $N\ge 0$ more than half of the integers in the set $\{0,1,\dots ,N\}$ are $P$‑representable. Prove that either every even intege... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 349192d2-88d6-5685-94de-4e970bff6a59 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For positive real numbers \(a,b,c,x,y,z\) satisfying \(a+x=b+y=c+z=1\), prove that
\[
(abc+xyz)\Big(\frac{1}{ay}+\frac{1}{bz}+\frac{1}{cx}\Big)\ge 3.
\] | {
"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 | 094ae2ce-bc84-5d73-889e-473ed983ee46 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | If \(a,b,c>0\), prove that
\[
\sqrt{\frac{(a^{2}+b^{2})(a^{2}+c^{2})}{b}}+
\sqrt{\frac{(b^{2}+c^{2})(b^{2}+a^{2})}{c}}+
\sqrt{\frac{(c^{2}+a^{2})(c^{2}+b^{2})}{a}}
\;\ge\;
\bigl(a^{2}+b^{2}+c^{2}+ab+bc+ca\bigr)\sqrt{\frac{3}{a+b+c}}\, .
\] | {
"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 | ee29ead4-328b-5467-97ee-16e63d752bb2 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let three equal circles $C_{1},C_{2},C_{3}$ be pairwise externally tangent and each internally tangent to a larger circle $C_{4}$. Let $S$ be a point inside $C_{4}$ from which tangents can be drawn to $C_{1},C_{2},C_{3}$, meeting them at $A,B,C$ respectively. Prove that one of the equalities
$$SA=SB+SC,\qquad SB=SC+SA... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | cf669408-5808-5229-b5a3-f4364c3d71c3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with incenter $I$ and circumcenter $O$. Let the lines $AI$, $BI$, $CI$ meet the circumcircle $(ABC)$ again at $P$, $Q$, $R$, respectively. Define
\[
X = BR\cap CQ,\qquad Y = CP\cap AR,\qquad Z = AQ\cap BP.
\]
Prove that the three circles $(XBC)$, $(YCA)$ and $(ZAB)$ 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 | 225ba72b-7c6d-596f-8e10-56ca4171c13b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that for every real number $M$ there exists an infinite arithmetic progression of positive integers whose common difference is not divisible by $10$ and such that the sum of the decimal digits of every term of the progression exceeds $M$. | {
"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 | 60f586a9-a233-58ca-bf44-c662c6ddf8f1 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ satisfy
\[
a+b+c=\frac{3\sqrt3}{\sqrt2}.
\]
Find the maximum value of
\[
S=\sum_{\text{cyc}}\frac{1}{a^{2}+b^{2}+3}
=\frac{1}{a^{2}+b^{2}+3}+\frac{1}{b^{2}+c^{2}+3}+\frac{1}{c^{2}+a^{2}+3}.
\] | {
"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 | 4a29f333-0b5a-5429-9eb3-3f3630e6e3d3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that for all positive real numbers \(a,b,c,x,y,z\),
\[
ax+by+cz+3\sqrt{(a^{2}+b^{2}+c^{2})(x^{2}+y^{2}+z^{2})}\;\ge\;4\sqrt{(ab+bc+ca)(xy+yz+zx)}.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 6705534e-84c2-50db-88ca-e1449932fe73 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For all positive real numbers $a,b,c$, prove the inequality
\[
\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge
\left(\frac{a+c}{b+c}\right)^{2}+\left(\frac{b+a}{c+a}\right)^{2}+\left(\frac{c+b}{a+b}\right)^{2}\,.
\] | {
"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 | b98ba459-34a0-5de1-b5c5-3b9b0a5f645e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying
\[f\bigl(x+f(x)f(y)\bigr)=f(x)+x\,f(y)\qquad\text{for all real }x,y.\] | {
"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 | b3c69153-713a-5ebf-9250-c358971a6bfe | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a$, $b$, $c$ be fixed positive integers. There are $a+b+c$ ducks sitting in a circle, one behind the other. Exactly $a$ ducks choose rock, $b$ ducks choose paper, and $c$ ducks choose scissors. A move consists of one of the following swaps:
<ul>
<li>If a rock‑duck sits directly behind a scissors‑duck, they exchang... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 69651bbd-1e9c-5f5d-97df-7cea8c8def5b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $P,Q\in\mathbb Z[x]$ be non‑zero polynomials with $\gcd(P,Q)=1$. For a positive integer $a$ define
$$X_a=\{\,n\in\mathbb Z\mid P(n)\equiv0\pmod a\,\}.$$
A pair of positive integers $(a,b)$ is called **sweet** if $X_a\neq\varnothing$ and for every $n\in X_a$ we have $b\mid Q(n)$.
Define
$$B=\{\,b\in\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 | caeb7f76-b78f-5047-bc4f-443a26ed1694 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Several clocks lie on a table. Let $X$ be a fixed point on the table. Assume that at some moment the sum of the distances from $X$ to the tips of the minute hands of all the clocks is not equal to the sum of the distances from $X$ to the tips of the hour hands. Prove that there exists a moment when the sum of the dista... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 4429edfb-6425-5578-90bc-1ade9139d9d6 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Solve the following equations for real $x$:
1. \(\displaystyle \sqrt{\,4+\sqrt{\,4+\sqrt{\,4-x\,}\,}\,}=x\).
2. \(\displaystyle x^{5}+x^{4}-12x^{3}-21x^{2}+x+5=0.\) | {
"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 | cfcd4b9e-90c7-52d9-a75b-c90af15bd608 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n$ be a given positive integer. Prove that there exist infinitely many integer triples $(x,y,z)$ such that
\[
n x^{2}+y^{3}=z^{4},\qquad \gcd(x,y)=\gcd(y,z)=\gcd(z,x)=1.
\] | {
"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 | 168bbfbf-e448-5e4e-8c92-563d24c0bce3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle. Let $BE$ and $CF$ be the altitudes from $B$ and $C$, intersecting $AC$ and $AB$ at $E$ and $F$ respectively, and let $H=BE\cap CF$ be the orthocenter. Let $A'$ be the reflection of $A$ in the line $BC$. Denote by $(ABC)$ the circumcircle of $ABC$, by $(AA'E)$ the circle through $A,\;A',\;E$, an... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 88f46319-50af-572b-9c12-603841c79e1d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute triangle. Let $B'$ and $C'$ be the reflections of $B$ and $C$ across the lines $AC$ and $AB$, respectively. Define $A_0 = B'C\cap BC'$. Let $D$, $E$, $F$ be the feet of the perpendiculars from $A_0$ to $BC$, $CA$, and $AB$, respectively, and let $I$ be the circumcenter of $\triangle DEF$. Prove 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 | 24c0dbf0-f4d3-5ac5-af0a-1fba12c9c74a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ with $a+b+c=3$. Prove that
\[
\sum_{\text{cyc}}\frac{a}{b^{2}+2}\ge 1.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | bea28d63-257d-59f0-84c9-39d81b8fc3d5 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(a,b,c\) be real numbers (not all zero). Prove that
\[
\frac{ab}{4a^{2}+b^{2}+4c^{2}}+\frac{bc}{4b^{2}+c^{2}+4a^{2}}+\frac{ca}{4c^{2}+a^{2}+4b^{2}}\le\frac13\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 0c369f5d-6d75-5f0a-8d46-1a2a88881ccd | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a cyclic quadrilateral and let $E=AC\cap BD$. Let $\omega$ be the circle tangent internally to the circumcircle $\Omega=(ABCD)$ at a point $T$ and tangent to the lines $EA$ and $EB$. Let $I$ be the incenter of $\triangle EAB$, let $N$ be the midpoint of the arc $AB$ of $\Omega$ that contains $C$, and let ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 1bc1792a-7054-5b26-97ab-5e91e7b7e6c0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $(O)$ be a circle and $AB$ a chord of $(O)$ that is not a diameter. The tangents to $(O)$ at $A$ and $B$ intersect at $C$. Let $M$ and $N$ be the midpoints of $AC$ and $BC$, respectively. Let $\omega$ be any circle that passes through $C$ and is tangent to $(O)$. Suppose $\omega$ meets the line $MN$ at two points $... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | b75bd3cf-bad6-5530-94a5-74f0fb657fbb | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $r$ and $s$ be two given circles and let $\ell$ be a given line. Using only ruler and compass, construct a circle whose centre lies on $\ell$ and which is tangent to both $r$ and $s$. | {
"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 | 3ebe4350-4d17-55f8-b26c-b89edb83af51 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine all functions $f : \mathbb{R}^+ \to \mathbb{R}^+$ such that
\[ f(xy+x^2)=f(x)f(y)+f(f^2(x)) \]
for all $x,y \in \mathbb{R}^+$. | {
"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 | 90663e20-4348-58bf-8a53-246a83078812 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that for all non‑negative real numbers $a,b,c$
\[
\sqrt{\frac{7a}{a+3b+3c}}+\sqrt{\frac{7b}{b+3c+3a}}+\sqrt{\frac{7c}{c+3a+3b}}\le 3.
\]
Equality holds if and only if $(a,b,c)$ is proportional to $(1,1,1)$ or to $(4,1,1)$ (up to permutation). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 9ca7575e-925a-565c-81b8-23a9b44a0130 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $I$ be the incenter of $\triangle ABC$. Let $D$, $E$, and $F$ be the reflections of $I$ across the lines $BC$, $CA$, and $AB$, respectively. Prove that there exists a point $P$ such that
$$AP\perp DP,\qquad BP\perp EP,\qquad CP\perp FP.$$ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 4bbebbf0-14f2-52bc-80ad-1b5bd0b75bf3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given a triangle $ABC$ with orthocenter $H$ and orthic triangle $DEF$, let $(N)$ be its nine‑point (Euler) circle. Choose a point $T$ on $(N)$. Let $D'$ be the reflection of $D$ across the line $AT$. Let $X$ be the intersection of the line through $D$ parallel to $AT$ with the line $D'T$. Let $(O_a)$ be the circumcircl... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 06b43690-f011-5c1e-a4f2-d72db94e41ba | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Decide whether there exist infinitely many triples \((a,b,c)\) of positive integers such that every prime divisor of \(a!+b!+c!\) is less than \(2020\). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | aba00ab6-4f5b-56e2-ab5b-e7a2243640de | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given a triangle $ABC$, let $AK$, $BL$, $CK$ be three cevians concurrent at a point $P$. Let $A'$, $B'$, $C'$ be the feet of the perpendiculars from $P$ to $BC$, $CA$, $AB$, respectively. Let $A''$, $B''$, $C''$ be the antipodal points of $A'$, $B'$, $C'$ on the circumcircle of $\triangle A'B'C'$ (i.e., $A''$, $B''$, $... | {
"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 | cea44fbd-ccef-5b3d-846d-6cf7a8889d67 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ such that $abc=1$. Prove that
$$\sum_{\text{cyc}}\frac{a\sqrt{b+c}}{b+c+1}\ge\sqrt2.$$ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 57ec1e74-8343-5988-91a4-73db67099f6e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that the set $\{\lfloor n\sqrt{2}\rfloor + \lfloor n\sqrt{3}\rfloor \mid n\in\mathbb{N}\}$ contains infinitely many even numbers and infinitely many odd numbers. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 33777fe0-983d-5488-9a35-94e993afe482 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be positive real numbers. Prove that
\[
\frac{a^{2}}{2a+5b}+\frac{b^{2}}{2b+5c}+\frac{c^{2}}{2c+5a}\ge\frac{3\,(a^{2}+b^{2}+c^{2})}{7\,(a+b+c)}\;.
\] | {
"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 | 767ea5e9-abe6-54aa-a572-5d771f4dd6dd | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all positive integers $x_1,x_2,\dots,x_{2004}$ such that
1) $x_i\ge 2\;$ for $i=1,2,\dots,2004$,
2) $\displaystyle \sum_{i=1}^{2004}x_i^{2}+4 \;=\;2002\cdot x_1x_2\cdots x_{2004}\,. $ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 3f7b8c85-e4f2-53f4-83a5-511632d76e0d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be positive real numbers and let $n>0$ be a positive real number. Prove that
\[
\sqrt[n]{a^{2}+2bc}+\sqrt[n]{b^{2}+2ca}+\sqrt[n]{c^{2}+2ab}\le
\sqrt[n]{a^{2}+b^{2}+c^{2}}+2\sqrt[n]{ab+bc+ca},
\]
\[
\sqrt[-n]{a^{2}+2bc}+\sqrt[-n]{b^{2}+2ca}+\sqrt[-n]{c^{2}+2ab}\ge
\sqrt[-n]{a^{2}+b^{2}+c^{2}}+2\sqrt[-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 | ca20cf02-88c3-5900-a20d-0fd2a19cf5c6 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute triangle with circumcircle $(O)$ and orthocenter $H$. Denote by $M,N,P$ the midpoints of $BC,CA,AB$, respectively, and let $D,E,F$ be the feet of the altitudes from $A,B,C$. Let $K$ be the reflection of $H$ across $BC$. The line $DE$ meets $MP$ at $X$, and the line $DF$ meets $MN$ at $Y$.
a) Le... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | e7f97888-8ac6-5843-a3a7-79c57fb29c66 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\{a_n\}_{n\ge0}$ and $\{b_n\}_{n\ge0}$ be two sequences of digits, i.e. $a_n,b_n\in\{0,1,\dots,9\}$ for every $n$. Assume there exists an integer $M$ such that $a_n\neq0$ and $b_n\neq0$ for all $n\ge M$. For each $n\ge0$ define the integer
\[
A_n=\overline{a_n a_{n-1}\dots a_1 a_0}=10^n a_n+10^{n-1}a_{n-1}+\cdots... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 43fbf12c-3c5d-5a80-a152-15b8d8e9ed8d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with $AB>BC$. Points $M$ on $AB$ and $N$ on $BC$ satisfy $AM=CN$. Let $K=MN\cap AC$. Let $P$ be the incenter of $\triangle KAM$ and let $Q$ be the $K$‑excenter (the excenter opposite $K$) of $\triangle KNC$. Prove that the midpoint of the arc $ABC$ of the circumcircle of $\triangle ABC$ is equid... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ce89fafe-7b6e-5495-83aa-737aaad8dec5 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | The centre of the incircle of triangle $ABC$ is $I$. Let $F$ be the midpoint of the circumcircle arc $BCA$ (the arc of $(ABC)$ that contains $C$). The line $FI$ meets the circumcircle again at $M$. Prove that the line $CM$ passes through the external centre of similitude (ex‑similicenter) of the incircle and the cir... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 9ccdad92-8fab-508e-86d6-b94d9b03f062 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For each positive integer $n$, let $f(n)$ denote the number of ways of representing $n$ as a sum of powers of $2$ with non‑negative integer exponents, where order of the summands does not matter. Prove that for every integer $n\ge 3$
\[
2^{\,n^{2}/4}<f\!\bigl(2^{n}\bigr)<2^{\,n^{2}/2}\,.
\] | {
"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 | 8a2fba8b-8e0b-506a-b220-cb5548225a3b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine all pairs of functions $f,g:\mathbb{R}\to\mathbb{R}$ such that
\[
f\bigl(g(y)+x\,f(y)\bigr)=g(xy)+f(y)
\]
holds for every 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 | 6cfc758e-75ed-53a9-bf50-194894ddac54 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all integer pairs \((a,b)\) with \(ab(a-b)\neq0\) for which there exists a subset \(Z_{0}\subseteq\mathbb Z\) such that for every integer \(n\) exactly one of the three integers \(n,\;n+a,\;n+b\) belongs to \(Z_{0}\). | {
"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 | 79679ba6-ff08-5cb4-a973-8e5b8295ae4c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that the only function $f:\mathbb{R}\to\mathbb{R}$ satisfying
(i) $\displaystyle\forall x\neq0,\; f(x)=x^{2}\,f\!\left(\frac1x\right)$,
(ii) $\displaystyle\forall x,y\in\mathbb{R},\; f(x+y)=f(x)+f(y)$,
(iii) $f(1)=1$
is $f(x)=x$ for all real $x$. | {
"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 | 626af9b0-988f-50b4-a5a0-8ca852b4c969 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that there is no integer $n$ such that $n^7+7$ 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 | 0e4311de-3c47-5540-85ef-bdb4404530ae | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a quadrilateral. Define $P=AB\cap CD$ and $Q=BC\cap AD$. Let $M$ be the Miquel point of $ABCD$, i.e. the common point of the four circles $(PAB),\;(PDC),\;(QAD),\;(QBC)$. Denote by $O_1,\;O_2,\;O_3,\;O_4$ the centers of these circles respectively. Prove that
1. $\triangle MO_1O_2\sim\triangle MAD$, 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 | ae6dc3c9-f8ae-5ad3-b070-8530fc363fd3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a cyclic quadrilateral whose four sides have pairwise distinct lengths. Let $O$ be the circumcenter of $ABCD$. The internal bisectors of $\angle ABC$ and $\angle ADC$ intersect $AC$ at $B_1$ and $D_1$, respectively. Let $O_B$ be the centre of the circle that passes through $B$ and is tangent to line $AC$ ... | {
"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 | 2bfa0f72-4d5b-5aae-85be-6b2ae4ecf07a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with incircle $\omega$. Let $D$ be a variable point on the circumcircle of $ABC$. Draw the two tangents from $D$ to $\omega$, and let $P$ and $Q$ be the feet of the perpendiculars from $A$ onto these two tangent lines. Prove that the line $PQ$ always passes through a point that does not depend o... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 57b43039-1ba2-5d9d-a514-808e72456d8e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | If $ABC$ is a triangle with side lengths $a,b,c$ and area $S$, prove that
\[
4\sqrt{3}\,S\;\ge\;\sqrt{(a^{2}+b^{2}+c^{2})\bigl[\,6(ab+bc+ca)-5(a^{2}+b^{2}+c^{2})\bigr]}\,.
\]
(Here $a,b,c>0$ and satisfy the triangle inequalities.) | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 42ae9309-da6a-54b3-9f7a-c3892b856337 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In triangle \(ABC\), let \(I\) be its incenter. Let \(N_a,\;N_b,\;N_c\) denote the nine‑point centers of the triangles \(BIC,\;CIA,\;AIB\), respectively. Prove that \(I\) is the orthocenter of triangle \(N_aN_bN_c\). | {
"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 | 51615269-4abb-587a-98c5-e0f9c9035d3e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In an acute triangle \(ABC\), let \(CF\) be the altitude from \(C\) to side \(AB\) (so \(F\in AB\)) and let \(BM\) be the median from \(B\) to side \(AC\) (so \(M\) is the midpoint of \(AC\)). Suppose that
\[
BM = CF\qquad\text{and}\qquad \angle MBC = \angle FCA.
\]
Prove that triangle \(ABC\) is equilateral. Additio... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a9194a3e-69e5-5487-9353-de8260ea8209 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For each positive integer $n$, let $f(n)$ denote the number of ways to represent $n$ as a sum of powers of $2$ (with non‑negative integer exponents), where the order of the summands does not matter. For example $f(4)=4$ because $4$, $2+2$, $2+1+1$, and $1+1+1+1$ are the four representations. Prove that for every intege... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 92cec21a-ce64-54dc-a45e-b774ae526829 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $p\ge 2$ be a prime. Eduardo and Fernando play a game taking turns, with Eduardo moving first. On each move the current player chooses an unused index $i\in\{0,1,2,\dots,p-1\}$ and then chooses a digit $a_i\in\{0,1,2,3,4,5,6,7,8,9\}$. After all $p$ indices have been chosen the number
\[
M=a_0+a_1\,10+a_2\,10^2+\... | {
"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 | ae072e99-f787-57a6-9855-d3c8610b164f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute triangle with orthocenter $H$. Let $B_1$ and $C_1$ be the feet of the altitudes from $B$ and $C$, respectively, and let $M$ be the midpoint of $AH$. Choose a point $K$ on the segment $B_1C_1$ such that $K\notin AH$. Let $AK$ intersect $MB_1$ at $E$ and $MC_1$ at $F$. Let $N=BE\cap CF$. 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 | fcc3f1db-8e55-5935-9344-6e56375f8fca | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | A balance has a left pan, a right pan, and a pointer that moves along a graduated ruler. If an object of weight $L$ is placed in the left pan and an object of weight $R$ in the right pan, the pointer stops at the number $R-L$ on the ruler. There are $n\;(n\ge 2)$ bags of coins, each bag containing $\displaystyle \frac{... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | be0d9352-47be-5164-817e-1bdc9c16a421 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let a circle S pass through the vertices A and C of a triangle ABC, and intersect the sides AB and BC in the points D and E, respectively (apart from A and C). Let F be the intersection of the lines CD and AE. The circumcircle of triangle BEF meets circle S again at a point G (different from E). Prove that the line GD ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 894c93f3-6a6d-59c1-817e-cd7caea52b7f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given positive real numbers $a,b,c$ satisfying $a+b+c=3abc$, prove that
\[
\frac{1}{a^2+b^2}+\frac{1}{b^2+c^2}+\frac{1}{c^2+a^2}+\frac{3}{2}\ge 2\Bigg(\frac{1}{a^2+bc}+\frac{1}{b^2+ca}+\frac{1}{c^2+ab}\Bigg).
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 2ac16889-1096-521f-8aa0-596566f87b9c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb R\to\mathbb R$ such that
$$f(xy-1)+f(x+y)=(f(x)+1)(f(y)+1)$$
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 | 87c6f2fc-826c-522b-bc17-053d665995b5 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\ge0$ with $a+b+c\neq0$. Prove that
\[
\sqrt{a^{2}-ab+b^{2}}+\sqrt{b^{2}-bc+c^{2}}+\sqrt{c^{2}-ca+a^{2}}
\le\frac{10(a^{2}+b^{2}+c^{2})-ab-ac-bc}{3(a+b+c)}.
\] | {
"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 | 4bf4c9d8-a8bf-50f2-b45d-424a6c45dc61 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with $O$ and $N$ the circumcenter and nine‑point‑circle center, respectively.
Denote by $O_a,\;O_b,\;O_c$ the circumcenters of triangles $OBC,\;OCA,\;OAB$, respectively, and by $N_a,\;N_b,\;N_c$ the nine‑point‑circle centers of triangles $OBC,\;OCA,\;OAB$, respectively.
Prove that the incent... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 88d7d2ae-12e8-542a-98e9-096c8287d350 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a$, $b$, $c$ be positive real numbers such that $ab+bc+ca\le 3abc$. Prove that
\[
\sqrt{\frac{a^{2}+b^{2}}{a+b}}+\sqrt{\frac{b^{2}+c^{2}}{b+c}}+\sqrt{\frac{c^{2}+a^{2}}{c+a}}+3\le \sqrt{2}\bigl(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\bigr).
\] | {
"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 | 39f9a4f7-e5d0-54f9-9fb8-0fafd7faa498 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with circumcircle $(ABC)$. Denote by $P$ the midpoint of the arc $AC$ not containing $B$, and by $Q$ the midpoint of the arc $AB$ not containing $C$. Let $I$ be the incenter of $\triangle ABC$. Choose an arbitrary point $M$ on the arc $BC$ of $(ABC)$ that does not contain $A$. Define $K=MQ\cap 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 | 4ecc5a9d-6056-5da9-bc91-2c9c60d3c4e0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCDEF$ be a convex hexagon such that $AB\parallel DE$, $BC\parallel EF$, and $CD\parallel FA$. Denote by $M,N,P,Q,R,S$ the midpoints of $AB$, $BC$, $CD$, $DE$, $EF$, and $FA$, respectively. Prove that the three lines $MQ$, $NR$, and $PS$ 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 | cb3e126a-d27e-5188-ad5e-f4234b0463c1 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | If $a,b,c\ge0$ and $a+b+c=3$, prove that
\[
(a^2+bc^4)(b^2+ca^4)(c^2+ab^4)\le 64.
\]
Equality holds for $(a,b,c)=(1,2,0)$ 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 | 0968fe69-2b4f-5986-b26a-e028a314d940 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $p$ and $q$ be two given positive integers with $\gcd(p,q)=1$.
A set of $p+q$ real numbers $a_1<a_2<\dots<a_{p+q}$ is called **balanced** if
* $a_1,a_2,\dots,a_p$ form an arithmetic progression with common difference $q$, and
* $a_p,a_{p+1},\dots,a_{p+q}$ form an arithmetic progression with common difference... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 49775862-c1e7-5bce-9e25-c0ac8edaa073 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with side lengths $BC=a$, $CA=b$, and $AB=c$. For any point $P$ in space, prove that
\[
\frac{PA\cdot PB}{ab}+\frac{PB\cdot PC}{bc}+\frac{PC\cdot PA}{ca}\ge 1.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 8f8d371e-30bf-5918-a828-ffe0694eadbb | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In an acute triangle $ABC$, let $K$ be a point on the extension of side $BC$. Choose points $P$ and $Q$ such that $KP\parallel AB$ and $BK=BP$, and $KQ\parallel AC$ and $CK=CQ$. Let the circumcircle of $\triangle KPQ$ meet the line $AK$ again at $T\neq K$. Prove that
1. $\displaystyle\angle BTC+\angle APB=\angle CQA... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | c9fcd69c-668c-5d3f-8a45-2cbf27facbe5 | [] | AoPS | Nemotron-Math-Proofs-v3 |
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