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 scalene acute triangle. Let $P$, $Q$, $R$ be the feet of the altitudes from $A$, $B$, $C$ onto $BC$, $CA$, $AB$, respectively. Let $D$, $E$, $F$ be the midpoints of $BC$, $CA$, $AB$, respectively.
From $P$ drop perpendiculars to the sides $DE$ and $DF$ of triangle $DEF$; let $X$ and $Y$ be the feet of... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | e52df59f-d6c9-5378-8ceb-cd72aa62c7be | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,y,z\ge0$. Prove each of the following four inequalities:
\[
\begin{aligned}
&(x-y)(x-z)(x+y)^2+(y-z)(y-x)(y+z)^2+(z-x)(z-y)(z+x)^2\ge0,\\[2mm]
&(x-y)(z-y)(x+y)^2+(y-z)(x-z)(y+z)^2+(z-x)(y-x)(z+x)^2\ge0,\\[2mm]
&(x-2y+z)(2x-y-z)(y+z)^2+(x+y-2z)(2y-x-z)(z+x)^2+(y-2x+z)(2z-x-y)(x+y)^2\ge0,\\[2mm]
&(x-2y+z)(2x-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 | 7785fb97-521d-5a78-8b2d-fccdb7019c63 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\triangle ABC$ be a triangle with incenter $I$, and let its incircle $\omega$ touch $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively. Let the line $AD$ meet $\omega$ again at $X\neq D$. Through $E$ and $F$ draw lines parallel to $AD$, meeting $\omega$ again at $P\neq E$ and $Q\neq F$ respectively. Prove that $I$ lie... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | e1325bca-0350-570c-b827-5b23d7567230 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,y,z\ge0$ satisfy $xyz(x+y+z)\le3$. Prove that
\[
5(x^2+y^2+z^2)+\frac{72}{xy+yz+zx}\ge39.
\]
Let $x,y,z\in\mathbb R$ satisfy $x+y+z=xy+yz+zx$ and $xyz\ge1$. Prove that
\[
(x^5-1)(y^5-1)(z^5-1)\ge0.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | d0d003c9-2809-5517-ba7a-ec7e334fdb43 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $B$, $C$, $D$ be the vertices of a triangle. Let $A$ be a point inside $\triangle BCD$ and let $E$ be a point on segment $BD$. Suppose $BC=BA$ and $\angle BCD=\angle CAE$. Let $\omega$ be the circumcircle of $\triangle EAD$, and let $F$ be the second intersection of $\omega$ with line $CD$ (different from $D$). Pro... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 41284530-87ec-56f5-8243-12aff721003e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that for every positive integer $n$,
\[
d(n)\,\bigl(n^{2}-n\bigr)\;\ge\; \bigl(\sigma(n)-n\bigr)\,\bigl(\sigma(n)-1\bigr),
\]
where $d(n)$ denotes the number of positive divisors of $n$ and $\sigma(n)$ (denoted $o(n)$ in the original post) denotes the sum of the positive divisors of $n$. | {
"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 | bdbffbaf-50cf-5bd6-a99c-7354d8ee9089 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with side lengths $a=BC$, $b=CA$, $c=AB$. Denote by $m_a$, $m_b$, $m_c$ the lengths of the medians from vertices $A$, $B$, $C$ to the opposite sides $a$, $b$, $c$, respectively. Prove that
\[
\frac{m_b\,m_c}{bc}+\frac{m_c\,m_a}{ca}+\frac{m_a\,m_b}{ab}\;\ge\;\frac94,
\]
and determine when 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 | 468fc55a-97ee-5e4e-a375-b517454b6656 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute triangle with orthocenter $H$. Define $E=BH\cap AC$ and $F=CH\cap AB$. Let $D$, $M$, $N$ be the midpoints of $AH$, $BD$, $CD$, respectively, and let $T=FM\cap EN$. Assuming the four points $D$, $E$, $T$, $F$ are concyclic, prove that the line $DT$ passes through the circumcenter $O$ of $\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 | 4915ee67-8c34-5680-b242-7b21bb78586b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb Z\to\mathbb Z$ such that for every pair of integers $x,y$ the integer $f(x)+f(y)+xy$ divides $xf(x)-y^3$, i.e.
\[f(x)+f(y)+xy\mid xf(x)-y^3,\]
where $a\mid b$ means there exists an integer $c$ with $ac=b$. | {
"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 | 808b9e8c-c7e4-58ae-836b-29dad50093b7 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n$ be a positive integer denoting the number of message bits. A binary code encodes each $n$‑bit message into a codeword of length $N=n+r$, where $r$ is the number of redundancy bits and $r<n$. Suppose the transmission channel can delete a single bit at an unknown position (i.e., the receiver obtains a word of l... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 3e58abae-8041-5335-8ae6-a341b4a1ceba | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all integer pairs $(x,y)$ satisfying $x^x = 1 + y^2$. | {
"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 | ca4f2c55-65f0-58a7-9f42-b8b6137d1361 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $I$ be the incenter of $\triangle ABC$. Let $\ell$ be the perpendicular bisector of $AI$. Let $P$ be any point on the circumcircle $(ABC)$. Define $Q=AP\cap\ell$. Let $R$ be the point on $\ell$ such that $\angle IPR=90^\circ$. Let $M$ be the intersection of the line $IQ$ with the midsegment of $\triangle ABC$ 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 | 61dd2804-d5d2-5397-a422-0ac08780345f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,y,z$ be positive real numbers. Prove that
\[
\frac{x^4}{8y^3+5z^3}+\frac{y^4}{8z^3+5x^3}+\frac{z^4}{8x^3+5y^3}\ge\frac{x+y+z}{13}\, .
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 03aa86d4-4d11-539b-bd23-c0c69054983d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be positive real numbers. Prove that
\[
\frac{a(b+c)}{\sqrt{b^{2}+bc+c^{2}}}\;+\;\frac{b(c+a)}{\sqrt{c^{2}+ca+a^{2}}}\;+\;\frac{c(a+b)}{\sqrt{a^{2}+ab+b^{2}}}\;\ge\;2\sqrt{ab+bc+ca}\,.
\] | {
"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 | 1cc60d1b-86d9-5091-9972-77f6eb78280e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a non‑isosceles acute triangle with altitudes $AD$, $BE$, $CF$ and orthocenter $H$ (so $D\in BC$, $E\in CA$, $F\in AB$).
Let the line $DE$ meet the circle with diameter $AD$ again at $M$, and the line $DF$ meet the same circle again at $N$.
Choose points $P\in AB$ and $Q\in AC$ such that $NP\perp AB$ ... | {
"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 | 4ab9f9ca-a5f1-5130-9eba-39b9490a0c2c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all real numbers $x,y$,
$$f\bigl(x+y\,f(x)\bigr)+f\bigl(x\,f(y)-y\bigr)=f(x)-f(y)+2xy.$$ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | e539085f-a504-52a5-af0c-a03dd75bf315 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c,d\in\mathbb{R}$ satisfy $ab=4$ and $c^{2}+4d^{2}=4$. Prove that
\[
(a-c)^{2}+(b-d)^{2}\ge\frac{8}{5}\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 842d84a3-8fb7-5103-a428-fab6554198ee | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In triangle $ABC$, let $X\in BC$, $Y\in CA$, $Z\in AB$ be points such that the cevians $AX$, $BY$, $CZ$ are concurrent. In triangle $XYZ$, let $D\in YZ$, $E\in ZX$, $F\in XY$ be points such that the cevians $XD$, $YE$, $ZF$ are concurrent. Prove that the three lines $AD$, $BE$, and $CF$ are concurrent (i.e., intersect ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 0e3dcddd-8bd4-5dfe-b8b3-1cf12293626b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Two circles intersect at points $A$ and $B$. Let $\ell$ be a line through $A$ that meets the first circle again at $C$ and the second circle again at $D$. On the first circle let $M$ be the midpoint of the arc $BC$ that does **not** contain $A$, and on the second circle let $N$ be the midpoint of the arc $BD$ that does... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 93122b26-72fd-551c-8d70-144e083ddd7d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $G$ be a circle and let $G_1,G_2,G_3,G_4$ be four circles each internally tangent to $G$. For $i,j\in\{1,2,3,4\}$ denote by $(i,j)$ the radical axis of $G_i$ and $G_j$.
1) Prove that the four radical axes $(1,2)$, $(2,3)$, $(3,4)$ and $(4,1)$ are concurrent if and only if the circles $G_1,G_2,G_3,G_4$ are external... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | f719226b-aeeb-5e7e-bac1-42453ee43f98 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that there exists an infinite sequence $(a_n)_{n=1}^{\infty}$ of integers greater than 1 such that each integer $>1$ appears exactly once in the sequence and for every $n\ge1$ the consecutive terms are not coprime, i.e. $\gcd(a_n,a_{n+1})>1$. | {
"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 | fb456db5-281c-5498-9791-0d6814722b61 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with circumcircle $\Omega$ and incenter $I$. Let the line through $I$ perpendicular to $BC$ intersect $AB$ and $AC$ at $X$ and $Y$, respectively. Let $P$ and $Q$ be points on $\Omega$ such that $\angle CPI = \angle BQI = 90^\circ$. Prove that the lines $PX$ and $QY$ intersect on $\Omega$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 839575c3-1203-5035-af96-79f68e71d6ca | [] | 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 $y$,
\[ f\bigl(x\cdot f(x+y)\bigr)+f\bigl(f(y)\cdot f(x+y)\bigr)=(x+y)^2. \] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 625c674a-72ca-50f0-9728-25e1791b5673 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given a triangle $ABC$, let $I$ be its incenter and $\Gamma$ its circumcircle. Let $D$ be the second intersection of line $AI$ with $\Gamma$. Choose a point $E$ on the arc $BDC$ of $\Gamma$ and a point $F$ on side $BC$ such that $\angle BAF=\angle CAE<\frac12\angle BAC$. Let $G$ be the midpoint of segment $IF$. Prove 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 | fe470648-5c7a-560c-8418-8ac659fa5abe | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | The incircle $\omega$ of triangle $ABC$ touches $BC, AC, AB$ at $D, E, F$ respectively. Two distinct points $K$ and $L$ are chosen on $\omega$ such that
\[
\angle CKE+\angle BKF=\angle CLE+\angle BLF=180^\circ .
\]
Prove that the line $KL$ is at equal distances from the points $D,\;E,$ and $F$ (i.e. $\operatorname{dis... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ca481a29-641d-57ca-bca7-b1ddf94d26ca | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all (finite) groups that have exactly six subgroups (including the trivial subgroup $\{e\}$ and the group itself). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 8e8d3717-91b9-59ca-8bb3-bb81fca92551 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\in\mathbb N$. If
\[
(a+\sqrt b)\,(b+\sqrt c)\,(c+\sqrt a)\in\mathbb Q,
\]
prove that
\[
(\sqrt a+b)\,(\sqrt b+c)\,(\sqrt c+a)\in\mathbb Q.
\]
(Here $\mathbb Q$ denotes the set of rational 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 | d308609b-1daf-5762-bc1c-e9bc9428c78c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\ge 0$ be real numbers such that no two of them are zero. Prove that
\[
\frac{a(4b+4c-a)}{b^{2}+c^{2}}+\frac{b(4c+4a-b)}{c^{2}+a^{2}}+\frac{c(4a+4b-c)}{a^{2}+b^{2}}\le\frac{21}{2}\, .
\] | {
"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 | d037100c-1b67-5cfb-a4b6-174ab3e6a2fe | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a right triangle with $AB<AC$. Let $AD$, $BE$, and $CF$ be the altitudes from $A$, $B$, and $C$ respectively, and let $H$ be the orthocenter. Denote by $M$ the midpoint of $BC$, and let $X$ be the second intersection of the circle with diameter $HM$ and the line $AM$. Let $T=HX\cap BC$. Prove that the 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 | 92ba3166-1b14-5991-82fe-491a833ee5db | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all integers $n$ (positive integers) such that the sum of the decimal digits of $n$, $2n$, $3n$, …, $n^2$ are all equal; i.e.
$$s(n)=s(2n)=s(3n)=\dots =s(n^2),$$
where $s(a)$ denotes the sum of the decimal digits of $a$. | {
"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 | 5f097abe-c10d-57b4-8578-fc4062be1a71 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\triangle ABC$ be a triangle with circumcircle $(O)$. Let the tangents to $(O)$ at $B$ and $C$ intersect at $D$. Let $E$ be the second intersection of $AD$ with $(O)$. Let $F$ be an arbitrary point on $(O)$, and define $G = BC\cap EF$. Let $M$ be the midpoint of $BC$. Prove that the points $A,M,F,G$ lie on a commo... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 1d02fbb6-12b7-5896-905a-fde110452749 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | A marker is placed at the origin of the integer lattice $\mathbb Z^2$. Calvin and Hobbes play a game: Calvin moves first and the players alternate turns. On each turn a player must choose two (not necessarily distinct) integers $a$ and $b$ that have never been chosen earlier by either player, and move the marker from $... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 4a4960c6-9e19-539c-9494-2613e6251960 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For positive real numbers $a,b,c$ satisfying $a^{2}+b^{2}+c^{2}+abc=4$, prove that
\[
a+b+c\le 3-\frac14\min\{(a-b)^{2},\,(b-c)^{2},\,(c-a)^{2}\}\,.
\] | {
"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 | f87d8bff-c860-5a07-b8ce-08b06047d6e9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with circumcenter $O$ and let $M$ be the midpoint of $BC$. Let $I$ and $J$ be the incenters of $\triangle ABM$ and $\triangle ACM$, respectively. The circumcircle of $\triangle AIJ$ meets the circumcircle $(O)$ again at $N$. Prove that $NO\perp BC$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 8735fca8-40f9-5939-9c43-712421654794 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle and let $BE$ be the internal bisector of $\angle ABC$. Let $D$ be a point on $BC$ such that $\angle DAC = \angle ABC$. Let $K$ be the incenter of triangle $ADC$. The line $EK$ meets the circumcircle of $ABE$ again at $L$. Prove that the circumcenter of triangle $LBC$ lies on the circumcircle of ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 3f8df247-7bdd-5ee8-8476-3e2eae43642d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | If $a,b,c$ are positive real numbers, prove that
\[
\frac{a}{\sqrt{4a^{2}+3ab+2b^{2}}}+\frac{b}{\sqrt{4b^{2}+3bc+2c^{2}}}+\frac{c}{\sqrt{4c^{2}+3ca+2a^{2}}}\le 1,
\]
\[
\frac{a}{\sqrt{4a^{2}+2ab+3b^{2}}}+\frac{b}{\sqrt{4b^{2}+2bc+3c^{2}}}+\frac{c}{\sqrt{4c^{2}+2ca+3a^{2}}}\le 1,
\]
\[
\frac{a}{\sqrt{4a^{2}+ab+4b^{... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 00228c7a-0d14-5308-96c2-15c6b0a5ec77 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let quadrilateral $ABCD$ be inscribed in a circle. Let $E$ be the intersection point of lines $AD$ and $BC$. Let $K$ and $L$ be the feet of the perpendiculars from $E$ to $AB$ and $CD$, respectively. Let $M$ and $N$ be the midpoints of the diagonals $AC$ and $BD$, respectively. Let $P$ be the intersection point of line... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | aadcd044-e4cb-5b25-980a-1fbf92580f14 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ such that
\[
\frac1a+\frac1b+\frac1c=\frac32 .
\]
Prove that
\[
\frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b}+3\;\ge\;
\sqrt{a^2+b^2+1}+\sqrt{b^2+c^2+1}+\sqrt{c^2+a^2+1}.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | f66f4bea-2dd6-5c14-ba26-678640dc39e5 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\ge0$ satisfy $(a+b)(b+c)(c+a)>0$. Prove that
\[
\frac{ab}{(a+b)^2}+\frac{bc}{(b+c)^2}+\frac{ca}{(c+a)^2}+\frac54\ge\frac{6(ab+bc+ca)}{(a+b+c)^2}\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 6d009760-0a73-5c24-8e9a-e26f2d5bee94 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all non‑negative integers $x,y$ such that $2\cdot3^x+1=7\cdot5^y$. | {
"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 | f28adc98-8218-55f7-8b22-ebb17c1a69db | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle. Let $P$ and $Q$ be isogonal conjugates with respect to $\triangle ABC$. Let $\ell_A$ be the line through $A$ perpendicular to $AP$; let $K=\ell_A\cap PB$ and $L=\ell_A\cap PC$. Let $\ell_Q$ be the line through $Q$ perpendicular to $BC$; let $D=\ell_Q\cap BC$, $M=\ell_Q\cap CA$, $N=\ell_Q\cap 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 | 1d78d164-a3ef-5c53-b043-7a5738c10293 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a scalene triangle with circumcircle $\Omega$, and let the incircle of $ABC$ touch $BC$ at $D$. The internal bisector of $\angle A$ meets $BC$ at $E$ and meets $\Omega$ again at $F$. Let $(DEF)$ be the circumcircle of $\triangle DEF$. The circle $(DEF)$ meets the $A$‑excircle at two points $S_1,S_2$ 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 | 786ec58a-e854-58bc-b946-e25ab3df92a9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions \(f:\mathbb{N}\to\mathbb{N}\) such that
\[
f(a+f(b))=f(a)^{\,f(b)}
\]
for all positive integers \(a,b\). | {
"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 | 9afd6b0e-4a94-5b5a-913b-c0ea49df8e3f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | A convex 2011‑gon is drawn. Peter draws its diagonals one by one, with the rule that each newly drawn diagonal may intersect (i.e., cross in the interior) at most one of the diagonals already drawn. (Diagonals that share an endpoint are not considered intersecting.) What is the greatest possible number of diagonals Pet... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 404a1a36-1934-51ac-a2f2-f78be6923dac | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For any positive integer \(n\), define
\[
S_n=\{\,x^{2}+ny^{2}\;:\;x,y\in\mathbb Z\,\}\subset\mathbb N .
\]
Find all positive integers \(n\) for which there exists an element of \(S_n\) that does **not** belong to any of the sets \(S_1,S_2,\dots ,S_{n-1}\). | {
"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 | 245f64a1-85d1-5718-981d-b28bc41a8e48 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a_{1},a_{2},\dots ,a_{2n+1}\in\mathbb{R}$ and let $n\ge 1$ be an integer. For each index $i$ with $1\le i\le 2n+1$ there exist two disjoint subsets $G_{1},G_{2}\subset\{1,2,\dots ,2n+1\}\setminus\{i\}$ such that
\[
|G_{1}|=|G_{2}|=n,\qquad
\sum_{j\in G_{1}}a_{j}=\sum_{j\in G_{2}}a_{j}.
\]
Prove that
\[
a_{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 | 920cfa57-f2d4-5d1b-9913-3f71b15e48e9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all positive integers $n$ such that $\displaystyle\frac{2^{\,n}+1}{n}$ is an integer (i.e. $n\mid 2^{\,n}+1$). | {
"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 | 499107a6-7161-55fe-9d9f-4f90d711e8ac | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with $AB>AC$ and let $I$ be its incenter. Let $M$ be the midpoint of the arc $BAC$ of the circumcircle $(ABC)$. Let $D=MI\cap BC$. Choose a point $E$ on $(ABC)$ such that the line $DM$ bisects $\angle BDE$. Let $F\neq E$ be the second intersection of the line $DE$ with $(ABC)$. Let $G=MF\cap BC$... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | fd80fde1-882b-5b87-88ee-17781f79ad2f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\ge 0$. Prove that
\[
4\bigl(a^2+b^2+c^2\bigr)+4k\,\sqrt[4]{a^3b^3c^3}+5k+12\ge (3k+8)(a+b+c),
\]
where $k=\dfrac{8}{\sqrt{19}+1}$. 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 | e35306b8-7dd3-5f8c-9822-401108f1f733 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $A,B,C,A',B',C'$ be six distinct points on a common circle. The lines $AA'$, $BB'$, $CC'$ intersect at a point $M$, and the acute angle between any two of these three lines is $60^\circ$. Prove that
(i) $AM+BM+CM = A'M + B'M + C'M$,
(ii) $\displaystyle \frac1{AM}+\frac1{BM}+\frac1{CM}= \frac1{A'M}+\frac1{B'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 | a486b30e-6030-5fee-9f80-6f273a3c521e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | 200 natural numbers $a_1,a_2,\dots ,a_{200}$ are placed in a row. For each $i=1,\dots ,199$ the adjacent numbers satisfy either $a_{i+1}=9a_i$ or $a_{i+1}=a_i/2$ (i.e., the right number is nine times the left number or one half of the left number). Determine whether it is possible that the sum $a_1+a_2+\cdots +a_{200}$... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | dcee2715-cb90-5504-9c9f-f3905bf86d37 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all real‑valued functions $f$ defined on a set of real numbers such that
\[
\frac{f(x+3)}{f(x+2)}=x+3
\]
holds for every $x$ where both $f(x+3)$ and $f(x+2)$ are defined (i.e. $f(x+2)\neq0$). | {
"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 | 152f69fc-ef59-58d5-ba0c-d2a699297234 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $C_1,C_2,\dots ,C_n$ be positive real numbers such that
\[
C_1C_2\cdots C_n=(n^2-1)^n .
\]
Prove that
\[
\frac1{\sqrt{1+C_1}}+\frac1{\sqrt{1+C_2}}+\cdots+\frac1{\sqrt{1+C_n}}\;\ge\;1 .
\]
(Here $n$ is a positive integer, $n\ge2$.) | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a5e2fdf8-42f1-5305-80bf-3ca2e8db1b04 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb R\to\mathbb R$ such that
\[
f(x+y)=f(x)+f(y)\qquad\text{for all }x,y\in\mathbb R,
\]
and
\[
f(x)\,f\!\bigl(\tfrac1{x}\bigr)=1\qquad\text{for all }x\neq0.
\] | {
"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 | fc03a53d-1c84-5ac7-bcca-8979b144cb5d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ be real numbers satisfying $a^{2}+b^{2}+c^{2}\le 3$. Prove that
\[
a^{2}+b^{2}+c^{2}+3abc\le 2(a+b+c).
\] | {
"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 | b43b5397-1d9f-57d4-a50f-9075c7df7ab3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Tom and Jerry play a game on a 25 × 25 chessboard. The board is initially empty. Starting with Tom, the players alternate placing a pawn on an empty square. After a move, if there exist four pawns that are the vertices of a rectangle whose sides are parallel to the sides of the board, the player who just moved wins. Wh... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a6ee2964-34da-527d-8972-b55151aa7dd4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Consider two $n\times n$ tables each containing the numbers $1,2,\dots,n^{2}$ exactly once (i.e., each table is a permutation of $\{1,\dots,n^{2}\}$).
Allowed operation: choose any $2\times2$ sub‑square of a table and rotate its four entries $90^\circ$ clockwise.
Find necessary and sufficient conditions on $n$ 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 | de63ddf8-0828-5156-9b04-2a72e271b7f5 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | A lattice point is a point in the plane with integer coordinates. Prove or disprove that there exists a finite set $S$ of lattice points such that for every line $l$ whose slope is $0$, $1$, $-1$, or undefined (i.e., a vertical line), the intersection $l\cap S$ is either empty or consists of exactly $2022$ 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 | 1788c697-05d9-5bf1-a8c1-8fe913c1b816 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle. Let $D$ be the second intersection of the internal angle bisector of $\angle BAC$ with the circumcircle of $ABC$. Let $I$ be the incenter of $ABC$ and let $M$ be the midpoint of $BC$. Let $P$ be the reflection of $I$ across $M$ (so $M$ is the midpoint of $IP$). The line $DP$ meets the circumcir... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 802925a3-1f67-558d-8618-f8e9c6509aa4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle. The medians of $ABC$ intersect its circumcircle again at points $A_1$, $B_1$, $C_1$, respectively. Prove that the perimeter of $\triangle A_1B_1C_1$ is not less than the perimeter of $\triangle ABC$, i.e. $P(A_1B_1C_1)\ge P(ABC)$. (Here $P(XYZ)$ denotes the perimeter of $\triangle XYZ$.) | {
"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 | c58eb127-6d2b-5630-833c-b0b3315bcd34 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all real numbers $x,y$,
\[
f(x+y)\,f\bigl(f(x)-y\bigr)=x\,f(x)-y\,f(y).
\] | {
"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 | de261e9a-95b7-55f6-a6b6-5a8da98a7ed4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Suppose $X$ is a set with $|X|=n$. A collection $A_1,\dots,A_k$ of non‑empty subsets of $X$ is called a **Sperner system** in $X$ if no $A_i$ is a subset of another $A_j$ (i.e. the family is an antichain).
Prove the following:
a) If $A_1,\dots,A_k$ is a Sperner system in $X$, then
\[
k\le \binom{n}{\left\lfloor 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 | 6d2a3586-c600-5c34-8f57-ef0ebc66413e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(ABC\) be a triangle and \(I\) its incenter. Let \(D\) be the point where the line through \(I\) perpendicular to \(AI\) meets side \(AC\). Let \(M\) be the midpoint of \(ID\) and \(N\) the midpoint of \(IB\). Let \(P\) be the midpoint of \(MN\). Prove that the line \(CM\) is tangent to the circumcircle of \(\{I,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 | b02539c8-8790-5c3a-97b4-f7850b491c32 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a\in(1,2)$ be a transcendental number.
1) Define $Y=\{P(a):P\in\mathbb Z[X]\}$. Show that $Y$ is dense in $\mathbb R$.
2) Define $X=\{P(a):P\in\mathbb Z[X]\text{ and every coefficient of }P\text{ lies in }\{-1,0,1\}\}$. Show that $X$ is dense 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 | 8e06edaf-ff8d-503b-acae-c1f0066fbd38 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n\ge k$ be positive integers and let $\mathcal{F}$ be a family of finite sets satisfying:
(i) $\mathcal{F}$ contains at least $\binom{n}{k}+1$ distinct sets each of size exactly $k$;
(ii) for any $A,B\in\mathcal{F}$ the union $A\cup B$ also belongs to $\mathcal{F}$.
Prove that $\mathcal{F}$ contains at least three... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 7e53a945-ca7f-5208-821f-9c8ad6113fb2 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that there are infinitely many pairs of consecutive positive integers that cannot be expressed as a sum of three squares, and prove that among any three consecutive positive integers there is at least one that can be expressed as a sum of three 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 | c9823dd7-0db0-50f5-bc3e-d065ef0e7369 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\triangle ABC$ be a scalene triangle with circumcenter $O$, incenter $I$, and incircle $\omega$. Let $\omega$ touch $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively. Let $T$ be the foot of the perpendicular from $D$ to the line $EF$. The line $AT$ meets the circumcircle of $\triangle ABC$ again at $X\neq A$. 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 | a8615164-478f-55c3-9142-dca3e46bfb9a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ satisfying each of the following equations for all $x,y>0$:
1. $f\bigl(f(x)+y\bigr)=x\,f(1+xy)$.
2. $f(x)\,f(y)=2\,f\bigl(x+yf(x)\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 | b1dc7ad2-b2a4-50fd-a010-551ea4eb6205 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\ge0$.
1. If
\[
\frac1{5a+1}+\frac1{5b+1}+\frac1{5c+1}=\frac12,
\]
prove that
\[
abc\ge1\qquad\text{and}\qquad\frac1{2a+1}+\frac1{2b+1}+\frac1{2c+1}\le1.
\]
2. If
\[
\frac1{5a+1}+\frac1{5b+1}+\frac1{5c+1}=1,
\]
prove that
\[
a^{2}+bc\ge\frac{12}{25}\qquad\text{and}\qquad\frac1{2a+1}+\frac1{2b+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 | ab04fe5f-4918-5d0c-b8d0-d6e2d8b75755 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ be real numbers such that $abc=1$. Prove that
\[
\frac{12a+7}{2a^{2}+1}+\frac{12b+7}{2b^{2}+1}+\frac{12c+7}{2c^{2}+1}\le 19.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a5bca5c3-5d18-56c0-ae95-6ca00fd3cd9e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Prove that
\[
ab+bc+ca+\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}\ge 2\bigl(a^{2}+b^{2}+c^{2}\bigr).
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 97577d4b-c2d2-574d-8da6-525ea10c4de8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit’s starting point $A_0$ and the hunter’s starting point $B_0$ coincide. After $n-1$ rounds of the game the rabbit is at point $A_{n-1}$ and the hunter is at point $B_{n-1}$. In the $n^{\text{th}}$ round three things occur in order:
1. The 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 | a172dff0-c748-5c25-a2a9-4d50d3062924 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In an acute scalene triangle $ABC$, let $D$, $E$, $F$ be the feet of the altitudes from $A$, $B$, $C$ onto $BC$, $CA$, $AB$, respectively. The altitudes $AD$, $BE$, $CF$ intersect at the orthocenter $H$. Points $P$ and $Q$ lie on the segment $EF$ such that $AP\perp EF$ and $HQ\perp EF$. Let $R$ be the intersection of l... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 46c09076-6384-52c1-a135-11bc33bd8bee | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a>k$ be natural numbers and let $r_1<r_2<\dots<r_n$, $s_1<s_2<\dots<s_n$ be strictly increasing sequences of natural numbers such that
\[
(a^{r_1}+k)(a^{r_2}+k)\cdots(a^{r_n}+k)=(a^{s_1}+k)(a^{s_2}+k)\cdots(a^{s_n}+k).
\]
Prove that the two sequences are identical, i.e. $r_i=s_i$ for all $i=1,\dots,n$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 592624e4-4958-5a1e-957a-51cb771bc995 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\Omega$ be the circumcircle of triangle $ABC$. Let $\omega$ be the circle tangent to sides $AC$ and $BC$ and internally tangent to $\Omega$ at point $P$. A line parallel to $AB$ that lies inside $\triangle ABC$ is tangent to $\omega$ at $Q$. Prove that
\[
\angle ACP=\angle QCB.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 565d8fb9-8931-5c6f-ae3c-e67641a4caa0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\triangle ABC$ be a triangle. Let $D$ be the midpoint of side $BC$. Let $E$ be the point on the circumcircle of $\triangle ABC$ that lies on the opposite side of line $AC$ from $B$ and satisfies $\angle DEA=\angle DEC$. Let $\omega$ be the circumcircle of $\triangle CED$. Let $X$ be the second intersection of $\om... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 095f5422-9318-5be9-85a7-fc066c695488 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute triangle. Denote by $D$ the foot of the perpendicular from $A$ to side $BC$, by $M$ the midpoint of $BC$, and by $H$ the orthocenter of $ABC$. Let $E$ be the intersection of the circumcircle $\Gamma$ of $ABC$ with the ray $MH$, and let $F$ be the second intersection of line $ED$ with $\Gamma$ (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 | 5af535a2-b35c-5772-ae21-a13e86223c2d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ such that $$f(x)\,f\!\Bigl(y+\frac{1}{f(x)}\Bigr)=f(xy+1)$$ 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 | 81c0cd70-7e2a-5751-b752-59d6c12d970a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given \(n\) lines in the plane in general position (no two are parallel and no three are concurrent), prove that there exist at least \(n-2\) triangles whose three sides lie on three of the given lines and such that the interior of each triangle contains no intersection point of any of the \(n\) lines. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | c5b785e4-d8e2-5e77-b307-4057cc47af41 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given a triangle $ABC$ with incenter $I$ and circumcenter $O$. Let $(BIC)$ denote the circumcircle of $\triangle BIC$ and let $E\neq C$ and $F\neq B$ be the second intersections of $(BIC)$ with $CA$ and $AB$, respectively. Let lines $BE$ and $CF$ meet the circumcircle $(O)$ again at $M\neq B$ and $N\neq C$. Let $K$ 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 | ef676be5-ed54-5505-9077-f873e0b674de | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $f:[0,1]\to\mathbb{R}$ be a function for which there exists a constant $K>0$ such that
\[
|f(x)-f(y)|\le K|x-y|\qquad\text{for all }x,y\in[0,1].
\]
Suppose also that for each rational number $r\in[0,1]$ there exist integers $a$ and $b$ such that $f(r)=a+br$. Prove that there exist finitely many intervals $I_1,\do... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 0b9a68f5-539e-5546-b365-d5c074d10074 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $r$ be a given circle in the plane, whose center is not marked. Using only a compass (i.e., you may draw circles with any chosen center and radius and find intersections of circles, but you may not draw straight lines), construct the center of $r$. | {
"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 | c9e3bb98-fa4c-532e-abb2-7c1255bea994 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$. Prove that
\[
a^{6}+b^{6}+c^{6}+4\sum_{\text{cyc}}a^{5}(b+c)\;\ge\;4\sum_{\text{cyc}}a^{3}b^{3}+15a^{2}b^{2}c^{2}.
\]
(Here $\displaystyle\sum_{\text{cyc}}a^{5}(b+c)=a^{5}(b+c)+b^{5}(c+a)+c^{5}(a+b)$ and $\displaystyle\sum_{\text{cyc}}a^{3}b^{3}=a^{3}b^{3}+b^{3}c^{3}+c^{3}a^{3}$.) | {
"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 | f2ca8221-14ca-513a-bba5-87082c4af239 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that
\[
x^{4}y+y^{4}z+z^{4}x+xyz\bigl(x^{3}+y^{3}+z^{3}\bigr)\ge (x+y+z)\bigl(3xyz-1\bigr)
\]
holds for all positive real numbers \(x,\;y,\;z\). | {
"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 | 27882a59-f210-50df-8a2e-6e9ff9bcfc77 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$. Prove that
\[
\frac{a^{2}+2bc}{(b+c)^{2}}+\frac{b^{2}+2ca}{(c+a)^{2}}+\frac{c^{2}+2ab}{(a+b)^{2}}\ge\frac94\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | e673ff41-f6be-5bcc-aef4-237e5f2bd56f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a non‑isosceles triangle with circumcenter $O$ and centroid $G$.
Denote by $A_1$, $B_1$, $C_1$ the feet of the altitudes from $A$, $B$, $C$ respectively.
Let $X$, $Y$, $Z$ be the second intersections of the rays $GA_1$, $GB_1$, $GC_1$ with the circumcircle $(O)$.
Let $D_1$ be the intersection of 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 | ddcf7d36-fe43-534b-ac0b-8a5ae251a895 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that every positive rational number $x$ can be expressed uniquely as a finite sum
\[
x = a_1 + \frac{a_2}{2!} + \frac{a_3}{3!} + \dots + \frac{a_n}{n!},
\]
where $n$ is a positive integer, $a_1$ is a non‑negative integer, and for each $k\ge 2$ the integer $a_k$ satisfies $0\le a_k\le k-1\,. $ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 147fa530-4d97-57a0-b3b1-fbd5c85b0abf | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\Omega$ be the circumcircle of $\triangle ABC$, and let $PQ$ be a diameter of $\Omega$ with $P,Q\in\Omega$. Let $\mathcal{H}$ be the rectangular circumhyperbola that is the isogonal conjugate of the line $PQ$. Prove that the asymptotes of $\mathcal{H}$ are the Wallace–Simson lines of $P$ and $Q$ with respect to $\... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 591ce6bd-85b5-5a75-84e2-1368b0447249 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n\in\mathbb N$. Prove that there are no natural numbers $x$ and $y$ such that
\[
\sqrt{n}+\sqrt{\,n+1\,}\;<\;\sqrt{x}+\sqrt{y}\;<\;\sqrt{4n+2}\,.
\] | {
"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 | d7c65fdb-209c-5cf4-8f22-bf0c258e6db0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute triangle with $AB<AC$. Let $D$ and $E$ be points on side $BC$ such that $BD=CE$ and $D$ lies between $B$ and $E$. Suppose there exists a point $P$ inside $\triangle ABC$ satisfying $PD\parallel AE$ and $\angle PAB=\angle EAC$. Prove that $\displaystyle\angle PBA=\angle PCA$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 6100cfdd-3866-5522-aa04-408a91f9275e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute‑angled triangle with orthocenter $H$ and circumcenter $O$. Let $D$ be the intersection of lines $AO$ and $BH$. Let $P$ be the point on side $AB$ such that $PH=PD$. Prove that the four points $B,\;D,\;O,\;P$ 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 | 044a4f89-e089-584c-a7b8-461faa0f2160 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $f_n(x)=\displaystyle\sum_{k=1}^{n}\frac{\cos(kx)}{k}$ for $x\in[0,\frac{\pi}{2}]$. Define $a_n=\displaystyle\min_{x\in[0,\frac{\pi}{2}]}f_n(x)$. Find $\displaystyle\inf_{n\ge 1}a_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 | 438c6f98-b1a4-534a-a75e-bb89455652c4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a fixed triangle. Let $l$ be a line passing through $A$ that does not intersect the side $BC$. Let $\omega_1$ be the circle with centre $O_1$ tangent to $BC$, to $l$, and to the side $AC$ (the circle lies outside $\triangle ABC$). Let $\omega_2$ be the circle with centre $O_2$ tangent to $BC$, to $l$, 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 | 8daa5888-e763-5656-8753-bd66591f78e8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that for all $a,b,c>0$,
\[
(a^2+b^2)\Bigl(\frac{2ab}{a+c}-c\Bigr)+(b^2+c^2)\Bigl(\frac{2bc}{a+b}-a\Bigr)+(c^2+a^2)\Bigl(\frac{2ca}{b+c}-b\Bigr)\ge0.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 4aaa1dcf-694a-581e-b2c6-44c73bb63d17 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine the smallest number of colors needed to color the edges of the complete graph $K_{2^n}$ (which has $2^n$ vertices) so that every Hamiltonian cycle in the graph contains at least $n$ distinct colors. Give a coloring that attains this minimum and prove that it satisfies the required property, and also prove tha... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 6cdd2400-bb6f-5efe-b747-f9507adec6c2 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n\ge1$ be an integer. Consider a $(2^n-1)\times(2^n-1)$ array $[a_{ij}]$ in which each entry $a_{ij}$ is either $1$ or $-1$. For each cell, its *neighbors* are the orthogonal adjacent cells that lie inside the array (up, down, left, right). Thus a corner cell has two neighbors, an edge (non‑corner) cell has thr... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 97471b36-771a-5918-96cd-7fb833b867bb | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a right triangle with $\angle C=90^\circ$. Let $D$ be the foot of the altitude from $C$ to the hypotenuse $AB$. Construct equilateral triangles $AED$ and $CFD$ outward on sides $AD$ and $CD$, respectively, such that $E$ lies on the same side of $AB$ as $C$ and $F$ lies on the same side of $CD$ as $B$. 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 | 0f6691ff-e8dc-5344-a1ff-9b392f8d831e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ satisfy $6(a^2+b^2+c^2)+10(ab+bc+ca)=3$. Prove that
\[
\frac{a}{(b+c)(b+2c)^2}+\frac{b}{(c+a)(c+2a)^2}+\frac{c}{(a+b)(a+2b)^2}\ge\frac{8}{3}\, .
\] | {
"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 | 55a6cef3-cf9d-544f-821e-986503f02b49 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For a real number $r$ let $f(r)$ denote the integer that is closest to $r$ (if the fractional part of $r$ is $1/2$, define $f(r)=r-\tfrac12$). Let $a>b>c$ be rational numbers such that for every integer $n$ the identity
\[
f(na)+f(nb)+f(nc)=n
\]
holds. Determine all possible triples $(a,b,c)$. | {
"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 | fb00b202-0e36-5410-a8c8-79f142059ba6 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with circumcircle $\odot(O)$, orthocenter $H$ and the Kosnita point $K$. Let $K_1$ be the reflection of $K$ in side $BC$. Let $D$ be the intersection of the tangents to $\odot(O)$ at $B$ and $C$. Prove that the points $H,\;K_1,\;D$ are collinear. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 22e12c2d-ed8b-5329-b6dc-5388b766d23e | [] | AoPS | Nemotron-Math-Proofs-v3 |
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