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"
} | In triangle $ABC$, let $BD$ be the altitude from $B$ to $AC$ and let $H$ be the orthocenter of $ABC$. Let the perpendicular bisector of $HD$ intersect the circumcircle of $BCD$ at points $P$ and $Q$. Prove that
\[
\angle APB+\angle AQB=180^{\circ}.
\] | {
"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 | 5c77c69b-2698-54ab-8042-11b7b380d6b3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $O$ and $H$ be the circumcenter and orthocenter of $\triangle ABC$, with altitudes $AD$, $BE$, $CF$. Let $(A,AD)$ denote the circle centred at $A$ with radius $AD$. The circles $(A,AD)$ and $(O)$ intersect at $M$ and $N$.
1. Prove that the line $MN$ bisects the segments $DE$ and $DF$.
2. Let $G=EF\cap BC$, 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 | 91efb55f-bfce-5928-82a9-212a0e0b4a20 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $(a_n)_{n\ge0}$ be a sequence of non‑negative real numbers such that $\displaystyle\sum_{n=0}^{\infty}a_n<\infty$.
Let $(u_n)_{n\ge0}$ be a real sequence satisfying
\[
u_n\ge u_{\,n-1}-a_{\,n-1}\qquad\text{for all }n\ge1,
\]
and suppose there exists a constant $M$ with $u_n\le M$ for all $n$. Prove that the 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 | 8a1cefcb-1287-5a41-9369-e8772af4fe23 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In triangle $ABC$ with incenter $I$, let $D$ be the second intersection of line $AI$ with the incircle $(I)$. Prove that the tangent line to $(I)$ at $D$ and the external angle bisector of $\angle BIC$ intersect at a point on side $BC$. | {
"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 | dbf8ef5f-6836-50f6-88c5-e799c80caaa1 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all real numbers $c$ for which there exists a function $f:\mathbb{R}\to\mathbb{R}$ satisfying
\[
(f(x)+1)(f(y)+1)=f(x+y)+f(xy+c)
\]
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 | 72ea6afd-7f29-502a-ba1a-d78b81b78b1d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(a,b,c\in[0,1]\) satisfy \(a+b+c=1\). Prove that
\[
a^{2}b+b^{2}c+c^{2}a+abc\le\frac{4}{27}.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 897bed0a-0e1a-5da4-9307-f5535c555d1e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a convex quadrilateral with $AC=BD$. Let $X=AC\cap BD$. Let $P\neq B$ be the second intersection of line $BC$ with the circumcircle of $\triangle AXB$, and let $Q\neq C$ be the second intersection of $BC$ with the circumcircle of $\triangle CXD$. On the circle $(AXB)$ denote by $R$ the midpoint of the arc... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 25a75453-a507-541d-9b01-9ea73b0236fb | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given points $A,B,C,D,E,F,G$ in the plane such that
$$AG = GC = CA = AF = DF = AD = EA = BE = AB,$$
and points $H,I,J$ satisfy
$$EH = HD,\qquad BI = IG,\qquad CJ = JF,$$
(i.e. $H$, $I$, $J$ are the midpoints of $ED$, $BG$, and $CF$ respectively), prove that the triangle $HIJ$ is equilateral; equivalently $H... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 0799af6a-1d2c-5225-bb27-5e8ac6e04aee | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute triangle. Prove that
$$\frac{h_a}{l_a}+\frac{h_b}{l_b}+\frac{h_c}{l_c}\ge \frac{21}{10}+\frac{9r}{5R},$$
where $h_a,h_b,h_c$ are the altitudes, $l_a,l_b,l_c$ are the internal angle‑bisector lengths, $r$ is the inradius and $R$ is the 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 | 1af42519-b882-5f2b-92a8-b36c10fb9e23 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute‑angled triangle with $AB<AC$, and let $H$ be its orthocenter. The circle with diameter $AH$ meets the circumcircle of $ABC$ again at $P\neq A$. The tangent to the circumcircle of $ABC$ at $P$ meets line $BC$ at $Q$. Prove that $QP=QH$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 5e6ef740-9cd8-51fc-8539-3f314f45b663 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | A polynomial $f(x)$ with integer coefficients is given. Define $d(a,k)=|f^{k}(a)-a|$, where $f^{k}$ denotes the $k$‑fold iterate ($f^{0}(x)=x$, $f^{k}(x)=f(f^{k-1}(x))$). It is known that for each integer $a$ and each natural number $k\ge1$, $d(a,k)>0$ (i.e. $f^{k}(a)\neq a$). Prove that for all such $a$ and $k$,
$$d(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 | 9955d1f2-9e40-54ee-963d-467cf4420d02 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$. Prove that
\[
a^{3}+b^{3}+c^{3}+3abc\;\ge\;ab\sqrt{2\!\left(a^{2}+b^{2}\right)}\;+\;bc\sqrt{2\!\left(b^{2}+c^{2}\right)}\;+\;ca\sqrt{2\!\left(c^{2}+a^{2}\right)}\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 49376b31-f556-5acb-b81d-780d3da3f132 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For a positive integer $n\ge 2$, define a sequence $a_1,a_2,\dots ,a_n$ by
\[
a_1=\frac{n(2n-1)(2n+1)}{3},\qquad
a_k=\frac{(n+k-1)(\,n-k+1\,)}{2(k-1)(2k+1)}\,a_{k-1}\qquad(k=2,3,\dots ,n).
\]
(a) Prove that each $a_k$ is an integer.
(b) Show that there is exactly one term among $a_1,\dots ,a_n$ that is not divis... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | b359ed8e-e268-5fbf-8caf-df4458a79881 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle and $P$ an interior point.
Let $\omega_A$ be the circle tangent to the circumcircle of $BPC$ at $P$ internally and tangent to the circumcircle of $ABC$ at $A_1$ internally.
Let $\Gamma_A$ be the circle tangent to the circumcircle of $BPC$ at $P$ externally and tangent to 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 | 94e0eee3-c468-5367-9d37-035d163b1107 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be the lengths of the sides of a triangle. Prove that
\[
(a+b+c)\Big(\frac{1}{b+c-a}+\frac{1}{c+a-b}+\frac{1}{a+b-c}\Big)\ge
3\Big(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\Big).
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 8a0b116a-6ae0-5a5c-b42e-3499c814e717 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given a circle \((O)\) and an inner circle \((O')\) that is internally tangent to \((O)\) at the point \(X\). Let \(A,B\) be arbitrary points on \((O)\). From \(B\) draw the tangent to \((O')\) touching \((O')\) at \(M\) (so \(BM\) is tangent to \((O')\)), and from \(A\) draw the tangent to \((O')\) touching \((O')\) 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 | 793b91f8-6592-5cf6-878b-af4a8039a15c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\triangle ABC$ with $AB<AC$ and let $I$ be its incenter. Let $(O)$ be the circumcircle of $\triangle ABC$. Denote by $M$ and $N$ the midpoints of the arcs $\widehat{BAC}$ and $\widehat{BC}$ of $(O)$, respectively. Let $D$ be the second intersection of the line through $A$ parallel to $BC$ with $(O)$, and let $E$ 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 | 67777bd1-f862-5ddc-9bb2-3c692eff8200 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | The function $f:\mathbb{N}\to\mathbb{N}$ is strictly increasing (i.e. $f(n+1)>f(n)$ for every $n\in\mathbb{N}$) and satisfies
\[
f\bigl(f(n)\bigr)=2n+2\qquad\text{for all }n\in\mathbb{N}.
\]
Determine the value of $f(2013)\,. $ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ca274081-ace3-5097-a8eb-d826a03a0292 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In triangle $ABC$, let $P$ and $Q$ be isogonal conjugates with respect to $ABC$. Let $O$ be the circumcenter of $ABC$. Let $H$ be the orthocenter of triangle $QBC$, and let $H'$ be the reflection of $H$ across line $BC$. Choose a point $D$ on the circumcircle of $ABC$ (the circle centered at $O$) such that $AD\perp PO$... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 1eeb3b0c-f98f-595e-88d7-5ea6c246a63e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For positive real numbers $a,b,c,d$ satisfying
$$a+b+c+d=\frac1a+\frac1b+\frac1c+\frac1d,$$
prove that
$$(a+b+c+d)(1+abcd)\ge 8.$$
Equality holds when $a=b=c=d=1$. | {
"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 | c7e70406-6536-5a31-a608-5df293fe5465 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | There are $n\ge 2$ line segments in the plane such that every two segments intersect and no three segments are concurrent. Geoff must choose one endpoint of each segment and place a frog on that endpoint, facing the other endpoint. After each of $n-1$ claps, every frog jumps forward along its segment to the next inters... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 8b234d0f-385d-5b76-8d8b-ac7b5fe88b64 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{Q}^{+}\to\mathbb{Q}$ such that
\[
f(x)+f(y)=\Bigl(f(x+y)+\frac1{x+y}\Bigr)\bigl(1-xy+f(xy)\bigr)
\]
for all $x,y\in\mathbb{Q}^{+}\,.$ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 77188f15-3dd1-530f-b339-4d6008b3009f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n\ge 3$ be a fixed integer. A deck of $3n$ distinct cards numbered $1,2,\dots,3n$ is shuffled and each of $n$ players sitting in a circle receives exactly three cards. In every turn each player simultaneously passes the smallest‑numbered card in his hand one place clockwise, passes the largest‑numbered card one pl... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | df93b04f-d447-539c-a2ac-0c2f740fe85d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying
\[
xf(x)+yf(xy)=xf\!\bigl(x+yf(y)\bigr)
\qquad\text{for all real }x,y.
\] | {
"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 | 21e55b91-f33f-5ab3-8b10-c29f27d970af | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Convex quadrilateral $ABCD$ is inscribed in a circle $\Omega$. The extensions of sides $AD$ and $BC$ intersect at $E$. Let $M$ and $N$ be the midpoints of the arcs $AB$ and $CD$ of $\Omega$ that do not contain the other vertices. Let $I,J,K,L$ be the incenters of triangles $ABD$, $ABC$, $BCD$, and $CDA$, respectively. ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 3ca3d832-9f02-5452-8c2e-d4c2c83cadc7 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle. Choose variable points $B_1$ on $BA$ and $C_1$ on $CA$ such that $BB_1=CC_1$. Let $B_2\neq B_1$ be the point on the circumcircle of $\triangle A C B_1$ for which $BC_1\parallel B_1B_2$, and let $C_2\neq C_1$ be the point on the circumcircle of $\triangle A B C_1$ for which $CB_1\parallel C_1C_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 | 17e301af-6bee-53f9-898d-8f69349c0c28 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c,d$ be positive real numbers. Prove that
\[
\sqrt{\frac{a^{3}}{a^{3}+15bcd}}+\sqrt{\frac{b^{3}}{b^{3}+15cad}}+\sqrt{\frac{c^{3}}{c^{3}+15abd}}+\sqrt{\frac{d^{3}}{d^{3}+15abc}}\ge 1.
\] | {
"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 | e6509704-202b-5a83-86c5-8ea24e9fb499 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Number $0$ is written on a board. Two players take turns adding symbols to the right end of the expression. On each turn the first player writes a sign $+$ or $-$, and then the second player writes a number from $\{1,2,\ldots,1993\}$ that has not been used before. After $1993$ such pairs have been written the expressio... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 83d878e4-8a46-57a5-9a5c-8fc346cde2fa | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\omega_1$ and $\omega_2$ be two circles intersecting at points $A$ and $E$. Let $\ell$ be the tangent to $\omega_2$ at $E$, and let $P$ be an arbitrary point on $\ell$. From $P$ draw the tangent $PD$ to $\omega_2$ (with $D\in\omega_2$) and the tangent $PB$ to $\omega_1$ (with $B\in\omega_1$). Let $T=AE\cap PB$. 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 | 3d664649-3435-56a4-a46e-653055766cd7 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In an acute-angled triangle $ABC$ with $\angle B>\angle C$, let $D$ be the foot of the altitude from $A$ to $BC$ and let $E$ be the foot of the perpendicular from $D$ to $AC$. Let $F$ be a point on segment $DE$. Prove that $AF\perp BF$ if and only if $EF\cdot DC=BD\cdot DE$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 91ab454e-6edd-5656-b026-4dcadef7cbc9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ and $n\in\mathbb N_0$. Define
\[
g_n(a,b)=\sum_{i=0}^{n}\frac{(a+1)_i\,(b)_{\,n-i}}{(c+1)_i\,(c)_{\,n-i}\;i!\;(n-i)!},
\]
where $(z)_m=z(z+1)\cdots(z+m-1)$ is the rising (shifted) factorial.
1. Prove that $g_n(a,b)>g_n(b,a)$ whenever $b>a$.
2. Since $b-a$ divides $g_n(a,b)-g_n(b,a)$, the quotien... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | d31f78a3-8db4-5422-a747-49579397c4e1 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | There are $n\ge2$ websites labeled $1,2,\dots,n$. For each $i\in\{1,\dots,n-1\}$ there is already a directed link from $i$ to $i+1$. We may add further directed links, but a link may only go from a smaller‑numbered website to a larger‑numbered website. Prove that we can add at most $3(n-1)\log_{2}\!\bigl(\log_{2}n\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 | e22c7f17-079f-52c1-b77a-836a0d32c713 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $f:\mathbb N\times\mathbb Z\to\mathbb Z$ be defined by
\[
f(0,0)=5^{2003},\qquad f(0,n)=0\ \text{for all }n\neq0,
\]
and for every integer $n$ and every natural $m>0$,
\[
f(m,n)=f(m-1,n)-2\Big\lfloor\frac{f(m-1,n)}2\Big\rfloor
+\Big\lfloor\frac{f(m-1,n-1)}2\Big\rfloor
+\Big\lfloor\frac{f(m-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 | 6b10af8e-ab23-5f0b-9e71-5c29333b7218 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that for non‑negative real numbers \(a,b,c\) (with \(ab+bc+ca>0\)) the following chain of inequalities holds:
\[
1+\frac{a^{2}+b^{2}+c^{2}}{ab+bc+ca}\;\ge\;
\sum_{cyc}\frac{a(b+c)}{a^{2}+2bc}\;\ge\;
1+\frac{ab+bc+ca}{a^{2}+b^{2}+c^{2}}.
\]
(The sum \(\displaystyle\sum_{cyc}\) denotes the cyclic sum over \(a,b,c\)... | {
"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 | 55f492c3-29a7-5e75-82e2-5176515ca78f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine whether there exists a quadruple \((x,y,z,w)\in\mathbb N^{4}\) such that
\[
\frac{x^{2}+y^{2}}{zw}=\frac{z^{2}+w^{2}}{xy}=137\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 3215bb41-c15b-5825-a1df-7bfe4645faca | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | A grid consists of all points of the form $(m,n)$ where $m$ and $n$ are integers with $|m|\le 2019$, $|n|\le 2019$ and $|m|+|n|<4038$. The points $(m,n)$ with either $|m|=2019$ or $|n|=2019$ are called **boundary points**; the four lines $x=\pm2019$ and $y=\pm2019$ are the **boundary lines**. Two points are **neighbo... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | f2ce7e57-3719-5df9-a3e9-d782a3f68a67 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that
\[
\int_{0}^{\infty}\frac{x^{4}+x^{2}+\frac{1}{12}}{\bigl(x^{4}+x^{2}+\frac{3}{4}\bigr)^{3}}\,e^{-x^{2}}\,dx=\frac{\sqrt{\pi}}{9}\, .
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 8e751ec7-e4c4-521f-8a94-501c5d8d5621 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In triangle $ABC$, let $M$ be the midpoint of $BC$ and let $D$ be a point on the segment $AM$. Distinct points $Y$ and $Z$ are chosen on the rays $\overrightarrow{CA}$ and $\overrightarrow{BA}$, respectively, such that
\[
\angle DYC = \angle DCB\qquad\text{and}\qquad \angle DBC = \angle DZB.
\]
Prove that the circumci... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | f50e61b5-0513-5c19-8d00-3698f066801b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given positive integers $m$ and $n$, prove that there is a positive integer $c$ such that the numbers $cm$ and $cn$ have the same number of occurrences of each non‑zero digit when written in base ten. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 28e416c8-3d8a-5dca-bb06-574d9d2387ce | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\Gamma_1$ be a circle and let $P$ be a point outside $\Gamma_1$. The two tangents from $P$ to $\Gamma_1$ touch the circle at $A$ and $B$. Let $M$ be the midpoint of segment $PA$, and let $\Gamma_2$ be the circumcircle of $\triangle PAB$. Let line $BM$ intersect $\Gamma_2$ again at $C\neq B$. Let line $CA$ intersec... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a8822260-fe3a-5f3c-b782-e7c66dbe4793 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a non‑degenerate triangle with circumcircle $\Gamma$ and incenter $I$. Let $M$ be the midpoint of side $BC$. Define points $D\in BC$, $E\in CA$, $F\in AB$ such that $ID\perp BC$, $IE\perp AI$, and $IF\perp AI$. Let $X\neq A$ be the second intersection of the circumcircle of $\triangle AEF$ with $\Gamma$. P... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ffb0090a-cd1c-5238-b90b-38b73c5c39e4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $N$ and $k$ be positive integers with $1\le k\le N$. Consider a simple undirected graph $G$ on $N$ vertices (the cities) in which each edge (airline) belongs to exactly one of $k$ companies (i.e. the edges are coloured with $k$ colours). The graph $G$ is connected, but for every company $c$ the subgraph obtained ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 4e5c9299-82b4-550f-a632-2dcc7acf4887 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Suppose $P(x)$ is a non‑constant polynomial with real coefficients and even degree. Bob writes $P(x)$ on a board. At each step, if the polynomial on the board is $f(x)$, he may replace it by
1. $f(x)+c$ for any real constant $c$, or
2. $P(f(x))$.
Can Bob always find a finite sequence of such steps that results ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 26681b69-0741-5d59-9fe8-3a1241e951ab | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | If $a,b,c,d$ are positive integers satisfying $abcd = a^{2}+b^{2}+c^{2}+1$, prove that $d = 4\,. $ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 07d8677f-e8c6-5621-bda9-41c74ed05315 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with circumcenter $O$ and incenter $I$. Let $A_0$, $B_0$, $C_0$ be the points where the incircle touches $BC$, $CA$, and $AB$, respectively. Let $(O_a)$ be the circle tangent to side $BC$ at $A_0$ and tangent to the circumcircle $(O)$ at $A_1$. Define $A_2=AA_1\cap BC$. Define $B_2$ and $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 | 5e268f07-e232-535b-a7b2-d170ed1d12b0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Each cell of a $1000\times1000$ table contains a $0$ or a $1$. Prove that one can either delete $990$ rows so that every column still contains at least one $1$, or delete $990$ columns so that every row still contains at least one $0$. | {
"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 | f835823c-d399-5d2b-b9d2-09f6ddf41a70 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ such that
\[
f(x)\,f(y)=f\bigl(x+y\,f(x)\bigr)\qquad\text{for all }x>0,\;y>0.
\] | {
"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 | 12499769-768d-57ad-9211-ca0dd86b56d9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb Z\to\mathbb N$ such that for every integers $m,n$, the number $f(2m+n)!\,+\,f(2n+m)$ is prime. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 9858abf7-5f3e-59a0-92ea-b5456eec2766 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For $a,b,c>0$ such that $a^3+b^3+c^3\ge a^2+b^2+c^2$, prove that
\[
\frac{a^3}{b(a+2b)}+\frac{b^3}{c(b+2c)}+\frac{c^3}{a(c+2a)}\ge 1.
\] | {
"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 | 3607edce-9e3a-5a59-b457-1cf556f3d9f2 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be the side lengths of a triangle (so $a,b,c>0$ and $a+b>c$, $b+c>a$, $c+a>b$).
1) Prove that
\[
(a+b+c)\Big(\frac1{a+b-c}+\frac1{b+c-a}+\frac1{c+a-b}\Big)\ge 9 .
\]
2) Prove that
\[
(ab+bc+ca)\Big(\frac1{(b+c-a)^{2}}+\frac1{(c+a-b)^{2}}+\frac1{(a+b-c)^{2}}\Big)\ge 9 .
\]
3) Find all real numbers $k$... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a0da792a-371f-53d5-ad4d-c74649be9660 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $G$ be a finite group. Let $H_1,\;H_2,\;H_3$ be proper subgroups (i.e. $H_i\neq G$) whose union equals $G$ (so $G=H_1\cup H_2\cup H_3$). Prove that for every $x\in G$, $x^{2}\in H_1\cap H_2\cap H_3$. | {
"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 | 5289a7ed-94ec-5322-a5a4-4f66432cbf65 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a convex quadrilateral inscribed in a circle $w$. Let the tangent to $w$ at $A$ intersect the lines $BD$, $BC$, and $CD$ at $X$, $Y$, and $Z$, respectively. Prove that the following three statements are equivalent:
1. $\displaystyle\frac{XY}{XZ}=\frac{AY}{AZ}$.
2. $AB\cdot CD=AD\cdot BC$.
3. $X$ ... | {
"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 | e77036b3-7991-5f50-99b4-cfd88184b2ee | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a cyclic quadrilateral. Let $E=AC\cap BD$ be the intersection of its diagonals and let $F=AB\cap CD$ be the intersection of the extensions of opposite sides. Let $K$ be the point such that $ABKC$ is a parallelogram (hence $AB\parallel CK$ and $BK\parallel AC$). Prove that $\displaystyle\angle AFE=\angle 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 | 68aac9af-9952-571e-8117-f8a3e664006a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,n$ be integers with $a>b>1$, $b$ odd, and $n\ge1$. If $b^n\mid a^n-1$, prove that
$$a^b>\frac{3^n}{n}\,. $$ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 00ac344b-9d0b-5051-87c0-b106c9048e51 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
\[
f\bigl(xf(x+y)+y^{2}\bigr)=yf(x+y)+xf(x)\qquad\forall\,x,y\in\mathbb{R}.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | d1a9d861-2e60-5d7e-a47a-e8081626cd8b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a cyclic quadrilateral. Let $M$ and $N$ be the midpoints of $AC$ and $BD$, respectively.
Define $P$ as the second intersection of the circumcircles of $\triangle ABM$ and $\triangle CDM$; define $Q$ as the second intersection of the circumcircles of $\triangle ABN$ and $\triangle CDN$; define $R$ as 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 | 0a2a7f94-b5e9-5388-b0e4-1e78b576daa4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given triangle \(ABC\) with circumcircle \((O)\) and incircle \((I)\). Let \((O_1), (O_2), (O_3)\) be circles each tangent to both \((O)\) and \((I)\), touching \((O)\) at \(A,B,C\) respectively and touching \((I)\) at \(A_1,B_1,C_1\) respectively. Let the incircle \((I)\) touch the sides \(BC,CA,AB\) at \(A_2,B_2,C_2\... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | dbefdb13-908c-51a0-b295-aff66f1fa21d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine all polynomials $f$ with integer coefficients such that for every odd prime $p$, $f(p)$ divides $2^{p}-2$. | {
"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 | 69e6413f-cf15-5764-8fa5-6314067617d9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with circumcircle $(O)$ and incircle $(I)$.
Let $D$ be the second intersection of the line $AI$ with $(O)$ (the point distinct from $A$).
Let $\ell$ be the line through $I$ perpendicular to $AD$, and let $M=\ell\cap BC$.
Let $H$ be the foot of the perpendicular from $M$ to the line $OI$ (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 | 554af608-4307-5c08-8bba-502e2e47929c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,y,z>0$. Prove that
\[
\sqrt{3+144\frac{x}{\,x+y\,}}+\sqrt{3+144\frac{y}{\,y+z\,}}+\sqrt{3+144\frac{z}{\,z+x\,}}\le 15\sqrt{3}\,.
\] | {
"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 | f755156c-92f8-50f8-876b-9e82a3e4e865 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n$ be a fixed positive integer. For each integer $k$ with $1\le k\le n$, let $S$ be any set of $k$ distinct prime numbers. Two players, Mary (who moves first) and Marco, alternately choose a positive integer greater than $1$ whose prime divisors all belong to $S$, with the rule that a chosen number may not be a mu... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 3f63f20d-aeb9-5f73-95cb-e52233c198ea | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Initially, two distinct positive integers $a$ and $b$ are written on a blackboard. At each step Andrea may choose two distinct numbers $x$ and $y$ already on the blackboard and write the number $\gcd(x,y)+\operatorname{lcm}(x,y)$ on the blackboard as well. Let $n$ be any positive integer. Prove that, regardless 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 | 8bb8797b-8f2e-512c-a059-d1f4a79e5818 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions \(f:\mathbb{R}^+ \to \mathbb{R}^+\) such that
\[
f\!\left(xy+\frac{f(y)}{y}\right)=f(x)f(y)+1
\]
for all real numbers \(x,y>0\). | {
"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 | e131067b-b1da-5249-882a-ac50622a484a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Solve in integers $n,m$ the equation
\[
1^2+2^2+\dots+n^2=m^2,
\]
i.e. find all integer pairs $(n,m)$ such that the sum of the first $n$ positive squares equals a perfect square. (Here $n$ is taken to be a non‑negative integer, so that the sum is defined.) | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ec82f0c1-5fb1-5a68-99da-02e92ad7b3a1 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(a,b,c>0\). Prove that
\[
\frac{a+b}{a^{2}+b^{2}}+\frac{b+c}{b^{2}+c^{2}}+\frac{c+a}{c^{2}+a^{2}}
\;\ge\;
\frac{3\,(a^{3}+b^{3}+c^{3})}{a^{4}+b^{4}+c^{4}}\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 022a8bd3-1846-58e5-942c-bf545cedbf49 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$. Let $E=AC\cap BD$ be the intersection of its diagonals. Let $P$ be a point inside $ABCD$. Let $O_{1},O_{2},O_{3},O_{4}$ be the circumcenters of $\triangle ABP$, $\triangle BCP$, $\triangle CDP$, and $\triangle DAP$, respectively. Prove that the three lines $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 | 287aceb2-ca1e-5d8b-a1a2-40c0488e1629 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with circumcircle $(O)$. Let $P$ be a point on line $BC$. Let $AP\cap(O)=\{A,P'\}$, where $P'\neq A$. Let $M$ be the midpoint of $BC$. The circumcircle of $P,P',M$, denoted $(PP'M)$, meets $(O)$ again at $N\neq P'$. Let $AN\cap(PP'M)=\{N,Q\}$, where $Q\neq N$. Let $I=PQ\cap AB$ and $J=PQ\cap AC$... | {
"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 | 5e18c8c0-99ac-5747-b9f4-f9309e1e938d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\ge0$. Prove that
\[
\sqrt{\frac{2a(b+c)}{(2b+c)(b+2c)}}+\sqrt{\frac{2b(c+a)}{(2c+a)(c+2a)}}+\sqrt{\frac{2c(a+b)}{(2a+b)(a+2b)}}\ge 2\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a4a847a2-51a7-5458-bb5b-ac6e409db226 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In an acute triangle $ABC$, let $D$, $E$, and $F$ be the feet of the altitudes from $A$, $B$, and $C$, respectively. Let $P\in DF$ and $Q\in EF$ be points such that $\displaystyle\angle PAQ=\angle DAC$. Prove that $AP$ bisects $\angle FPQ$ (i.e. $\displaystyle\angle FPA=\angle APQ$). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 0dcc2d9c-f57a-5151-85d4-c53136e3f433 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb N\cup\{0\}\to\mathbb N\cup\{0\}$ such that $f(1)=1$ and
\[f(m^2+n^2)=f(m)^2+f(n)^2\]
for all $m,n\in\mathbb N\cup\{0\}$. | {
"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 | ea8530b7-7a22-58f4-b032-71f3988ef3cd | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\triangle ABD$ and $\triangle ACE$ be right triangles with $\angle ADB=\angle AEC=90^\circ$, and suppose the two triangles are similar (equivalently $\angle BAD=\angle EAC$). Let $F$ be the intersection of the lines $BE$ and $CD$. Prove that $AF\perp DE\,. $ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | f1167382-c601-51dd-95a7-ef3faef8b9d9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | The incircle of triangle $ABC$ has centre $I$ and touches $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Let $P$ be a point on the same side of $FE$ as $A$ such that $\angle PFE = \angle BCA$ and $\angle PEF = \angle ABC$. Prove that $P$, $I$, and $D$ 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 | 27dc3c9f-e4ab-5d56-aedd-4830101bc003 | [] | 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 real $x,y$:
1. $f(x^2+f(y))=\dfrac{y}{2}+2f(x)^2$.
2. $f(2f(x+y))=f(x+y)+2f(x)f(y)-\dfrac{xy}{2}$.
3. $f(x+y)+y\le f\bigl(f(f(x))\bigr)$.
4. $x+f(xy+f(y))=f\bigl(f(x)\bigr)+yf(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 | 60c4ebb4-c0e9-520a-bbf7-81f76a13a0dd | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine all functions $f:\mathbb N\to\mathbb N$ such that for all natural numbers $m,n$,
$$f(n)+mn+f(m)\;\mid\;f(n)^2+f(n)f(m)+f(m)^2.$$ | {
"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 | c71cbe23-deff-5309-9499-d6794d1bcf25 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all integer pairs \((x,y)\) satisfying each of the following Diophantine equations:
(a) \(x^3+1=y^2\),
(b) \(3x^2-3x+1=y^2\). | {
"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 | 10f4c9a2-c2d7-5ad3-bb73-d98d971cb3f5 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCDEF$ be a convex hexagon with interior angles satisfying
\[
\angle A=\angle E=180^\circ-\tfrac12\angle C\qquad\text{and}\qquad
\angle B=\angle D=180^\circ-\tfrac12\angle F.
\]
Prove that the triangle formed by the perpendicular bisectors of $FA$, $AB$, and $BC$ is congruent to the triangle formed by the perpe... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 59a6084c-78ba-57dc-b15e-6d70bdf2fdaf | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For any integer $d>0$, let $f(d)$ denote the smallest positive integer that has exactly $d$ positive divisors (e.g. $f(1)=1,\;f(5)=16,\;f(6)=12$). Prove that for every integer $k\ge 0$ the number $f(2^k)$ divides $f(2^{k+1})$. | {
"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 | c0a97456-8dfb-55fe-b6c2-fd15dbee48ba | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | The tangent at $C$ to $\Omega$, the circumcircle of a scalene triangle $ABC$, meets $AB$ at $D$. Through $D$ a line meets $AC$ at $K$ and $BC$ at $L$. On $AB$ choose points $M$ and $N$ such that $AC\parallel NL$ and $BC\parallel KM$. Let $P=NL\cap KM$, which lies inside $\triangle ABC$. Let $\omega$ be the circumcircle... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ef1d2b19-123a-56a1-83a9-e0bad46e5f80 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a convex quadrilateral. Points $E$ on $AB$ and $F$ on $CD$ satisfy
\[
\frac{AB}{AE}=\frac{CD}{DF}=n,
\]
with $n>1$. If $S$ denotes the area of quadrilateral $AEFD$, prove that
\[
S\;\le\;\frac{AB\cdot CD\;+\;n(n-1)\,AD^{2}\;+\;n\,AD\cdot BC}{2n^{2}}.
\] | {
"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 | 56428d0f-4f86-5655-af8e-5935e429f0f4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ satisfy $a+b+c+2=abc$. Prove that
\[
\frac{\sqrt{1+ab+ac}}{a+1}+\frac{\sqrt{1+bc+ba}}{b+1}+\frac{\sqrt{1+ca+cb}}{c+1}\ge 3,
\]
and
\[
\frac{\sqrt{1+b^2+c^2}}{a+1}+\frac{\sqrt{1+c^2+a^2}}{b+1}+\frac{\sqrt{1+a^2+b^2}}{c+1}\ge 3.
\] | {
"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 | cebf8e30-f164-550d-93c2-c13957f3eef3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In triangle $ABC$, let $AD$ be the internal bisector of $\angle A$ meeting $BC$ at $D$. Let $AH$, $BK$, $CL$ be the three altitudes, with $H\in BC$, $K\in CA$, $L\in AB$ the foot points. From $D$ drop perpendiculars to $AC$ and $AB$, meeting them at $E$ and $F$, respectively. Let $M$ be the foot of the perpendicular fr... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 5c0b1cef-8822-5fb8-b62b-3b5d4c63f2e5 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute scalene triangle. Let $D$ be a point on the $A$‑altitude of $ABC$. The circle with diameter $AD$ meets $AC$, $AB$, and the circumcircle $(ABC)$ again at $E$, $F$, and $G$, respectively. Let $O$ be the circumcenter of $ABC$ and let $AO$ intersect $EF$ at $T$. The circumcircles $(ABC)$ and $(GTO)$ i... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 8f4b8d3e-ba87-5ad8-80bf-040b2b6aef51 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(0)$ is rational and
\[
f\bigl(x+f(y)^2\bigr)=\bigl(f(x+y)\bigr)^2\qquad\text{for all real }x,y.
\] | {
"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 | 6c07340d-53d3-5d07-abe6-13f6948360cd | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $I$ be the incenter of $\triangle ABC$ with $AB>AC$, and let $AI$ intersect $BC$ at $D$. Let $P$ be a point on $BC$ such that $PI=PD$. Let $J$ be the reflection of $I$ across the perpendicular bisector of $BC$. Let $Q$ be the second intersection of the circumcircles of $\triangle ABC$ and $\triangle APD$. 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 | 1772be9a-1d3d-5cfa-9823-d8086840f04b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle. Let $D$ be a point in the interior of $ABC$ such that $\angle ABD=\angle ACD$. Denote by $(I_1)$ and $(I_2)$ the incircles of $\triangle ABD$ and $\triangle ACD$, respectively. Apart from the line $AD$, these two incircles have a second internal common tangent. Prove that this tangent always pa... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 02964f18-6717-5e75-9840-fe3214249274 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $P(x)$ be a monic polynomial with real coefficients satisfying
\[
P(x^2+x)=\bigl(P(x)\bigr)^2+P(x)\qquad\text{for all }x\in\mathbb R.
\]
Prove or disprove that every such polynomial is obtained by the recursion
\[
P_1(x)=x,\qquad P_{n+1}(x)=P_n(x^2+x)\ \ (n\ge1).
\]
(Aside: the only non‑monic polynomial sol... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 80137061-4a8c-5ad7-b179-1e3405213148 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\triangle ABC$ be any triangle. Construct points $A_0$, $B_0$, $C_0$ in the plane such that
\[
\angle A_0BC=\frac{C}{2},\qquad \angle A_0CB=\frac{B}{2},
\]
and define $B_0$, $C_0$ cyclically (i.e. $\angle B_0CA=\frac{A}{2},\ \angle B_0AC=\frac{C}{2}$ and $\angle C_0AB=\frac{B}{2},\ \angle C_0BA=\frac{A}{2}$). Pr... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | c0c73568-0eb9-556b-bdaf-e3a8486cc169 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle and let $X\in BC$, $Y\in CA$, $Z\in AB$ be points lying in the interior of the respective sides. Let $G$ be the centroid of triangle $XYZ$. Consider the homothety with centre $G$ and factor $4$, which maps $X,Y,Z$ to $X',Y',Z'$ respectively. Prove that triangle $X'Y'Z'$ contains (covers) at leas... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 0e2adbd4-9f73-5f7a-acc8-c46fc531187b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In triangle $ABC$, let $P$ be a point on side $AB$ and $Q$ a point on side $AC$ such that $PQ\parallel BC$. Prove that the circles with diameters $BQ$ and $CP$ intersect at points lying on the line through $A$ that is perpendicular to $BC$ (i.e., the altitude from $A$). | {
"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 | b9c680aa-aa5b-5299-bfe3-60e361a24ac0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute‑angled triangle with circumcircle $\Omega$. Let the internal angle bisectors of $\angle B$ and $\angle C$ meet $\Omega$ again at $M$ and $N$, respectively. Let $I$ be the intersection of these two bisectors (the incenter). Let $M'$ be the reflection of $M$ across side $AC$, and $N'$ the reflection... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 5417da2a-2b02-54dc-9963-07df550c5f72 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute triangle and let $AA_{1}$, $BB_{1}$, $CC_{1}$ be its altitudes (so $A_{1}\in BC$, $B_{1}\in CA$, $C_{1}\in AB$).
Let $(O)$ be the circumcircle of $ABC$.
Define $C_{2}$ as the point on the minor arc $AB$ (the arc of $(O)$ not containing $C$) such that the circle through $A_{1}$ and $B_{1}$ is 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 | 8d2851f6-c861-539e-bd0b-fe2fc55e4243 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given three sequences $(a_1,\dots,a_6)$, $(b_1,\dots,b_6)$, $(c_1,\dots,c_6)$, each of which is a permutation of the numbers $1,2,3,4,5,6$, find the minimum and maximum possible values of
\[
A=\sum_{i=1}^{6} a_i\,b_i\,c_i .
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | fe798775-d86b-5374-abd7-f6829e26595c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | We consider a prism whose top and bottom faces are pentagons with vertices $A_1,A_2,A_3,A_4,A_5$ and $B_1,B_2,B_3,B_4,B_5$, respectively. Every edge of the two pentagonal faces (the ten edges $A_iA_{i+1}$ and $B_iB_{i+1}$, indices taken modulo $5$) and every segment $A_iB_j$ for $i,j\in\{1,\dots,5\}$ is colored either ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 59c3e7ca-2b93-516e-9e64-f471c2aa8596 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\in\mathbb R$ satisfy $a+b+c=3$. Prove that
\[
\frac{1}{2+a^{2}+b^{2}}+\frac{1}{2+b^{2}+c^{2}}+\frac{1}{2+c^{2}+a^{2}}\le\frac34\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 84c1fe8b-a1f8-5aa8-b2bc-a6f96d6fb5f3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\in\mathbb R$. Prove that
\[
\bigl(a^{2}+b^{2}+c^{2}\bigr)^{3}\ge 2\,(a-b)^{2}(b-c)^{2}(c-a)^{2}\,.
\]
Equality holds iff $a,b,c$ are in arithmetic progression (i.e. $b-a=c-b$). | {
"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 | fe6995cb-f55f-5d7d-9f4d-6f42bb857aea | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute triangle with $AB<AC$. Let $AD$, $BE$, $CF$ be the altitudes, meeting $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively. Let $P=BC\cap EF$. Through $D$ draw the line parallel to $EF$, and let this line meet $AB$, $AC$, $CF$ at $Q$, $R$, $S$, respectively. Prove that
a) $B,Q,C,R$ 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 | 0b14a558-1771-544f-bc38-7550e713ad68 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | If $x,y,z$ are real numbers such that no two of them are zero, prove the following inequalities:
\[
(a)\qquad \frac{8x^{2}-5yz}{y^{2}-yz+z^{2}}+\frac{8y^{2}-5zx}{z^{2}-zx+x^{2}}+\frac{8z^{2}-5xy}{x^{2}-xy+y^{2}}\ge 9,
\]
\[
(b)\qquad \frac{5x^{2}+2yz}{2y^{2}+3yz+2z^{2}}+\frac{5y^{2}+2zx}{2z^{2}+3zx+2x^{2}}+\frac{5z^{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 | 88f74bb2-2545-57b6-bffd-53049600564b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute triangle. Let $\omega_{BC}$ be the circle having $BC$ as its diameter. Let $\omega_A$ be the circle tangent to the lines $AB$ and $AC$ and tangent externally to $\omega_{BC}$ at $A_1$. Define $B_1$ and $C_1$ analogously (i.e. $\omega_B$ is tangent to $BA$ and $BC$ and externally tangent to 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 | df27552a-533a-5ae7-9943-80167104c457 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that $(2^m-1)(3^n-1)$ cannot be a perfect square for any positive integers $m$ and $n$. | {
"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 | 2bc9237b-eb46-5a62-9a85-7c405ee0f6a3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For some integer $n$, can the set of consecutive integers $\{n,\;n+1,\;n+2,\;\dots,\;n+1997\}$ be partitioned into (at least two) subsets such that the product of the elements in each subset is the same? | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | d6ab3104-2b69-5b9d-b2b5-d0cbf22b950d | [] | AoPS | Nemotron-Math-Proofs-v3 |
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