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 $k$ be a positive real number. Let $ABC$ be an acute, non‑isosceles triangle with circumcenter $O$. Let $AD$, $BE$, $CF$ be the internal angle bisectors, meeting the opposite sides at $D$, $E$, $F$. On the rays $AD$, $BE$, $CF$ choose points $L$, $M$, $N$ such that
\[
\frac{AL}{AD}=\frac{BM}{BE}=\frac{CN}{CF}=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 | f751b451-d3e4-5695-8fb1-b608ca3970f7 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}^+\to\mathbb{R}$ satisfying
\[
f(x)+f(y)=f\bigl(x^{\,y}+y^{\,x}\bigr)
\qquad\text{for all }x,y>0.
\] | {
"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 | 19888d98-3c98-526d-b224-359a1415e689 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\triangle ABC$ be a non‑degenerate triangle with incenter $I$ and circumcenter $O$.
Define
* $M_A$, $M_B$, $M_C$ as the circumcenters of $\triangle BIC$, $\triangle CIA$, $\triangle AIB$, respectively;
* $I_A$, $I_B$, $I_C$ as the excenters of $\triangle ABC$ opposite $A$, $B$, $C$, respectively;
* $S_9$ ... | {
"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 | ac274f4a-a6e3-5ea5-ba45-e92c05edfdab | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a convex quadrilateral. Its interior angles at $B$ and $C$ satisfy
\[
\angle B+\angle C=\frac{2\pi}{3}\;(=120^\circ).
\]
Points $E$ and $F$ lie on side $BC$ such that $BE=EF=FC$ (i.e. $E,F$ trisect $BC$).
Moreover $\angle ABC=2\angle BAE$.
Prove that $ABCD$ is tangential (has an incircle) and 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 | bfd6f43b-91f2-5f35-83f5-42d0fd28a367 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In $\triangle ABC$ assume $\angle A>\angle B>\angle C$. Let $O$ be the circumcenter of $\triangle ABC$ and $I$ be the incenter. Let $D$ be the midpoint of $BC$. Let $M$ be the midpoint of the arc $BC$ of the circumcircle that does not contain $A$. Let $F$ be the point on line $ID$ such that $D$ is the midpoint of $IF$ ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | e7e550fa-7f8e-59d9-9547-3b0aa2eba081 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In an isosceles trapezoid $ABCD$ with $AD\parallel BC$, $AD=7$, $AB=CD=15$, $BC=25$, let $E$ be a point on $AB$ such that $\frac{AE}{BE}=1/2$. Let $F$ be a point on $BC$ and denote $BF=x$. Let $G$ be the intersection of line $EG$ with side $BC$ and let $H$ be the intersection of line $EG$ with line $CA$. The point $F$ ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 8e81cac0-8a5b-5cff-a97b-5a37d1991065 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Consider a circle $C_1$ and a point $O$ on it. Let $C_2$ be the circle with centre $O$ that intersects $C_1$ at two points $P$ and $Q$. Let $C_3$ be a circle externally tangent to $C_2$ at $R$ and internally tangent to $C_1$ at $S$, and assume that the line $RS$ passes through $Q$. Let $X$ (respectively $Y$) be the sec... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | e2d25f7a-02cd-5992-bcbe-b5dbd16dbd29 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In triangle $ABC$, let $M$ be the midpoint of side $BC$.
Let $\omega$ be a circle lying inside $\triangle ABC$ and tangent to $AB$ at $E$ and to $AC$ at $F$.
The two tangents drawn from $M$ to $\omega$ intersect $\omega$ again at points $P$ and $Q$, with $P$ on the same side of line $AM$ as $B$.
Define $X=PM\cap ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ab389e5e-5c2a-513e-bc75-ae91da3f3d5a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
\[f\big(x+f(f(y))\big)+f\big(xf(y)\big)=f(x)-y-f(xy)\]
for all real numbers $x,y$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 2b5a1ff4-0a39-58e5-8f29-175864224518 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that the Diophantine equation $x^2 - p\,y^2 = -1$ has infinitely many integer solutions when $p$ is a prime of the form $4k+1$, and has no integer solutions when $p$ is a prime of the form $4k+3$. | {
"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 | 4372986d-2300-535e-951e-3a287d640c68 | [] | 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
\[
\sqrt{2a^{2}-a+1}\;+\;\sqrt{2b^{2}-b+1}\;+\;\sqrt{2c^{2}-c+1}\;\le\;\sqrt{5\!\left(a^{2}+b^{2}+c^{2}\right)+3}.
\]
Equality occurs when $a=b=c=1\,. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 73f4fcb7-4dbd-5436-80c6-9c47637366de | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Seventeen football fans plan to attend 17 distinct World Cup matches. Each fan may purchase at most one ticket for any given match. For any two different fans, the set of matches for which they both have tickets contains at most one match (i.e., any two fans share at most one common match). It is known that there is 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 | 73fec742-1e82-5142-8c59-c69ff7fd4ef3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute triangle with circumcenter $O$. Choose points $E\in OB$ and $F\in OC$ such that $BE=OF$. Let $M$ be the midpoint of the arc $EOA$ of the circumcircle of $ABC$, and let $N$ be the midpoint of the arc $AOF$ of the same circle. Prove that
\[
\angle ENO+\angle OMF=2\angle BAC.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 851f3e17-9ccf-562d-9c34-bf1dceae46bd | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given non‑negative real numbers $a,b,c$ (with $a+b>0$, $b+c>0$, $c+a>0$), find the minimum value of
\[
A=\frac{a^{2}+2bc}{(b+c)^{2}}+\frac{b^{2}+2ac}{(a+c)^{2}}+\frac{c^{2}+2ab}{(a+b)^{2}}\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 419a23bd-df52-57ca-aca0-c610b0ef217a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$. Prove that
\[
\sum_{cyc}\Bigg(\frac{a}{a+b}+\frac{b}{b+c}\Bigg)\Bigg(\frac{b}{b+c}+\frac{c}{c+a}\Bigg)\le
\frac{(a+b+c)^2}{ab+bc+ca}\, .
\] | {
"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 | f6499f77-b6ac-5ab3-a80c-10802432cb67 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ such that
\[f(xf(x))\,f(y)=f(x+y)\]
for all $x,y\in\mathbb{R}^{+}$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 09e2553a-ce94-5018-9cec-1d18fe8bf696 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$. Prove the following three inequalities.
1. \[
\sqrt{\frac{a^{2}+2b^{2}}{a^{2}+ab+bc}}
+\sqrt{\frac{b^{2}+2c^{2}}{b^{2}+bc+ca}}
+\sqrt{\frac{c^{2}+2a^{2}}{c^{2}+ca+ab}}\ge 3.
\]
2. If $a^{2}+b^{2}+c^{2}=1$, then
\[
\Bigl(a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2}\Bigr)
\left(\frac1{(1-ab)^{2}}+\frac1{(1-bc)^{2}}+... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | abc1788b-88d7-5757-995f-f61b42edf8ca | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x_1,x_2,x_3,x_4\ge0$. Prove that
\[
\bigl(x_1^2x_2+x_2^2x_3+x_3^2x_4+x_4^2x_1\bigr)
\bigl(x_1x_2^2+x_2x_3^2+x_3x_4^2+x_4x_1^2\bigr)
\ge x_1x_2x_3x_4\,(x_1+x_2+x_3+x_4)^2.
\] | {
"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 | 13dd922a-1777-56fd-bf33-b40956883984 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be non‑negative real numbers. Prove that
\[
a^3+b^3+c^3+3abc\;\ge\;ab\sqrt{2\bigl(a^2+b^2\bigr)}+bc\sqrt{2\bigl(b^2+c^2\bigr)}+ca\sqrt{2\bigl(c^2+a^2\bigr)}\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | f4689d97-12a2-597b-a702-3cf29e7c49de | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Unit cubes are made into beads by drilling a hole through each cube along a space diagonal. The beads are placed on a string so that they can move freely in space, with the restriction that the vertices of two neighboring cubes must touch. Let $A$ be the beginning vertex and $B$ the end vertex of the string. Suppose th... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 8b604d8a-b646-50d9-8c2a-b54b2c3a7d9f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with incenter $I$.
Let $I_a,\ I_b,\ I_c$ be the reflections of $I$ in the sides $BC,\ CA,\ AB$, respectively.
Define $(O_{ab})$ as the circumcircle of $\triangle AB I_b$ and $(O_{ac})$ as the circumcircle of $\triangle AC I_c$.
Let $R_a$ be the radical axis of circles $(O_{ab})$ and $(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 | c0536f6e-6982-5bf4-89e6-f2d07a73930e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n$ be a perfect number and let $k$ denote the number of its positive divisors. Prove that $$n<2^{2^k}$$. | {
"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 | 29ff012a-3b84-5c19-ba0d-9175dc52878d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | The quadrilateral $ABCD$ is inscribed in a circle. Let $E=AB\cap CD$ (the extensions of opposite sides) and $F=AC\cap BD$ (the intersection of the diagonals). The circumcircles of $\triangle AFD$ and $\triangle BFC$ intersect again at a point $H\neq F$. Prove that $\displaystyle \angle EHF=90^{\circ}\,.$ | {
"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 | 05b7cca0-771e-5bdc-8d6f-32a7a714fc08 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute triangle, i.e. $\max\{A,B,C\}<90^\circ$. Prove that
\[
\frac{\cos A\cos B}{\sin2C}+\frac{\cos B\cos C}{\sin2A}+\frac{\cos C\cos A}{\sin2B}\ge\frac{\sqrt3}{2}.
\]
Equality holds for $A=B=C=60^\circ$. | {
"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 | 283a4dbe-775b-539b-9d3f-60fa6cd35f8d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute triangle with circumcircle $(O)$. Let $B'$ be the second intersection of line $AC$ with $(O)$ and $C'$ the second intersection of line $AB$ with $(O)$. Define
\[
P = BB'\cap CC',\qquad
D = AP\cap (O)\ (\text{the second intersection of }AP\text{ with }(O)),\qquad
E = DB'\cap (O)\ (\text{the seco... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 6681052f-ca4a-5d9f-84a9-feb385b89d42 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $(O)$ be the circumcircle of $\triangle ABC$. Let $P$ and $Q$ be isogonal conjugates with respect to $\triangle ABC$. Let $D$ be the foot of the perpendicular from $Q$ to $BC$, and let $E$ and $F$ be the feet of the perpendiculars from $P$ to $CA$ and $AB$, respectively. Let $H$ be the orthocenter of $\triangle DEF... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 3af044a6-cd1b-5cd9-be96-fc15728045b8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $m,n$ be positive integers with $m\ge 2$ and $n < \frac{3}{2}(m-1)$. Consider a multigraph with $m$ vertices (cities) and $n$ edges (roads), where each edge joins two distinct vertices and multiple edges between a pair of vertices are allowed. Prove that the vertices can be partitioned into two nonempty sets $\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 | 201905d8-7649-5ec1-a813-e9ac7b2da302 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | There are 30 students in a class. Their examination scores are all distinct. Every student has the same number of friends, i.e. the friendship relation forms a simple undirected regular graph on 30 vertices. A student is called “nice” if his score is higher than the scores of more than half of his friends (strict major... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | feb9a423-df2f-5da5-a83c-a533d5874458 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | There are 2022 equally spaced points on a circular track γ of circumference 2022. The points are labeled $A_1,A_2,\dots,A_{2022}$ in some order, each label used exactly once. Starting at $A_1$, Bunbun the bunny hops to $A_2$, then to $A_3$, and so on until $A_{2022}$, after which she hops back to $A_1$. For each hop 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 | 5c2aec8c-0483-513a-8865-1ce919d74396 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given a triangle $ABC$ with circumcircle $\omega$ and incenter $I$. Let $N_A$ be the intersection of the external angle bisector of $\angle A$ with $\omega$. The line $IN_A$ meets the circumcircle of $\triangle IBC$ again at $T_A$. Define $T_B$ and $T_C$ analogously using the external bisectors at $B$ and $C$. Prove th... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ab96b5c8-aa7a-530b-a630-144157ccfc75 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $p$ be an odd prime. An integer $a$ with $1\le a<p$ is called a **quadratic non‑residue modulo $p$** if the congruence $a\equiv t^{2}\pmod p$ has no solution $t\in\mathbb Z$.
Define
\[
A=\{\,a\in\{1,2,\dots ,p-1\}\mid a\text{ and }4-a\text{ are both quadratic non‑residues modulo }p\,\}.
\]
Compute the remaind... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 12dc961b-231b-52c5-9450-06e065ce21cd | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(ABC\) be a non‑isosceles triangle. Denote by \(I\) its incenter and by \(O\) its circumcenter. On side \(BA\) choose a point \(A_{1}\) and on side \(CA\) choose a point \(A_{2}\) such that
\[
BA_{1}=CA_{2}=BC.
\]
Let \(O_{a}\) be the circumcenter of \(\triangle AA_{1}A_{2}\). Define points \(B_{1},B_{2},C_{1},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 | 4456ed26-baba-59a6-acc0-5324142e4974 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\triangle ABC$ have circumcenter $O$ and incenter $I$. Points $D$, $E$, $F$ lie on sides $BC$, $CA$, $AB$ respectively such that
\[
BD+BF=CA\qquad\text{and}\qquad CD+CE=AB.
\]
Let $P\neq D$ be the second intersection of the circumcircles $(BFD)$ and $(CDE)$. Prove that $OP=OI$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | dbc0b3e7-566d-5093-9895-f95f99940f10 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In the coordinate plane there are finitely many disjoint line segments (called walls), and no wall is parallel to either the x‑axis or the y‑axis. A bulldozer starts at an arbitrary point and initially moves east (in the +x direction). Whenever it reaches a point of a wall, it makes a right‑angle turn, away from the wa... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | fc80ab8e-1de0-53d1-be21-1efc347bf032 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(S=\{x\in\mathbb R\mid x>-1\}\). Find all functions \(f:S\to S\) such that
(a) \[f\bigl(x+f(y)+x\,f(y)\bigr)=y+f(x)+y\,f(x)\qquad\forall\,x,y\in S,\]
(b) the ratio \(\dfrac{f(x)}{x}\) is strictly increasing on each of the intervals \((-1,0)\) and \((0,\infty)\). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ad0e5eb8-cea7-594f-9a4a-2dccc076e16c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $O$ be the circumcenter and $I$ the incenter of $\triangle ABC$, and let $\Gamma$ be its circumcircle. Let $M$ and $N$ be the midpoints of the arcs $\widehat{AC}$ and $\widehat{AB}$ of $\Gamma$ that do **not** contain $B$ and $C$, respectively. Choose the point $P\in\Gamma$ such that $AP\parallel MN$. Let $T$ be th... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 65373b6c-89f5-5f0e-880c-12b8969f36df | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with incircle $(O)$.
Let $AD$ be the altitude from $A$ to $BC$ (so $AD\perp BC$).
Let $E=AO\cap BC$.
Let $F$ be the intersection of the two tangents drawn from $B$ and $C$ to $(O)$.
Let $P\neq A$ be the second intersection of line $AF$ with $(O)$.
Prove that the incircle $(O)$ is tange... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | f8c537e5-774a-5533-aeaa-830fd7426473 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with circumcircle $\omega$. Points $D$ and $E$ lie on sides $AB$ and $AC$, respectively, and are distinct from the vertices. Let the internal bisector of $\angle BAC$ meet $\omega$ again at $A'$ (with $A'\neq A$).
Define $P\neq D$ as the second intersection of line $A'D$ with $\omega$, and $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 | f8a8ef7b-01ad-5cb5-baae-45940f4561c0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
$$f(xy+f(x))=xf(y)+f(x)$$
for all real numbers $x$ and $y$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 6153407d-3e6c-5500-9d46-541e07d63c6b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $A_1A_2\ldots A_n$ be a convex $n$‑gon.
For a point $P$ inside the polygon denote by $P_i$ the foot of the perpendicular from $P$ onto the side $A_iA_{i+1}$ (indices are taken modulo $n$, so $A_{n+1}=A_1$).
Let $X_i\;(i=1,\dots ,n)$ be arbitrary points on the sides $A_iA_{i+1}$ respectively.
Prove that
\[
... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ec35199b-274b-53cf-a66f-8378fac1d6fa | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying each of the following functional equations.
1. $f(x)f(y)+f(xy)=f(2xy)$ for all real $x,y$.
2. $f(xy-1)=f(x)f(y)-1$ 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 | f71ae90f-a691-58d6-b300-012cdb17f96e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $C>1$ be a real constant. A sequence of positive real numbers $(a_n)_{n\ge1}$ satisfies $a_1=1$, $a_2=2$, the multiplicative property
\[
a_{mn}=a_m\,a_n\qquad\text{for all positive integers }m,n,
\]
and the additive bound
\[
a_{m+n}\le C\bigl(a_m+a_n\bigr)\qquad\text{for all positive integers }m,n.
\]
Prove th... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 618d5cd5-3185-585f-95e9-ab52532df490 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ with $abc=1$. Prove that
$$a^{2}+b^{2}+c^{2}+3\ge ab+bc+ca+a+b+c.$$ | {
"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 | 472ed96e-be33-5b01-94f6-4450e261bfc8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $P$ be a point inside convex quadrilateral $ABCD$. Points $Q_{1}$ and $Q_{2}$ are interior to $ABCD$ and satisfy
\[
\angle Q_{1}BC=\angle ABP,\qquad \angle Q_{1}CB=\angle DCP,
\]
\[
\angle Q_{2}AD=\angle BAP,\qquad \angle Q_{2}DA=\angle CDP .
\]
Prove that
\[
\overline{Q_{1}Q_{2}}\parallel\overline{AB}\ \Lon... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | b100561f-ed8c-5e0a-ad34-7b951d0c030c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all integers $m$ such that $m^2+7$ is a perfect cube. Equivalently, determine all integer solutions $(x,m)\in\mathbb Z^2$ of the Diophantine equation
\[
x^3=m^2+7.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 41a46c8c-d3af-5371-b190-6552a77fd504 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In triangle $ABC$, let $V$ be the reflection of the incenter $I$ across the circumcenter $O$. Let $H$ be the orthocenter and let $H'$ be its reflection across $O$. The line $BH'$ meets $AC$ again at $X$. Let $D$ be the $A$‑intouch point (the point where the incircle touches $BC$). Let $P=AB\cap ID$, $Q=BV\cap IX$, 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 | bd6b71a1-98b7-57d1-89b5-94c56090611a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n$ be a positive integer. A real polynomial $P$ is called **$n$‑pretty** if the equation
\[
P(\lfloor x\rfloor)=\lfloor P(x)\rfloor
\]
has exactly $n$ real solutions (here $\lfloor x\rfloor$ denotes the greatest integer $\le x$). Prove that for every positive integer $n$
a) there exists an $n$‑pretty polynomi... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 9d70af51-840b-5ee3-9afe-7b6f2e36d95f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given triangle $ABC$, let $I_a$ be its $A$‑excenter and let $D$, $E$, $F$ be the points where the $A$‑excircle $(I_a)$ touches $BC$, $CA$, and $AB$, respectively. Define
$$M=I_aB\cap DE,\qquad N=I_aC\cap DF,\qquad S=FM\cap EN.$$
Prove that the line $I_aS$ passes through the circumcenter $O$ of triangle $ABC$ (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 | ffd6bc31-8d7d-5117-b652-e5e6cb8ec4bb | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | A quadrilateral $ABCD$ is inscribed in a circle $(O)$. Let $P$ be the intersection of the internal angle bisectors of $\angle DAB$ and $\angle ABC$, and let $Q$ be the intersection of the internal angle bisectors of $\angle BCD$ and $\angle CDA$. Let $M$ be the midpoint of the arc $BC$ of $(O)$ that does not contain $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 | 5df63f9e-d6b4-57b3-8243-1c8e4b4657c6 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ with $abc=1$. Prove that
\[
\frac{3a^{2}+2b^{2}}{a\,(a^{2}+b^{2})}+\frac{3b^{2}+2c^{2}}{b\,(b^{2}+c^{2})}+\frac{3c^{2}+2a^{2}}{c\,(c^{2}+a^{2})}\ge\frac{15}{2}.
\] | {
"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 | 08f48b0c-515f-5e98-97d5-32919822c1f8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Positive integers $1,2,\dots ,n$ ($n>2$) are written on a blackboard. Each minute two numbers are erased and the least prime divisor (i.e., the smallest prime factor) of their sum is written on the board. The process continues until only one number remains, which must be $97$. Determine the smallest $n$ for which this ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 5b7ec32a-4988-56ef-ad04-809bb7993b56 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given a non‑right triangle $ABC$ with $BC>AC>AB$. Let $P_1\neq P_2$ be two points in the plane such that for each $i=1,2$ the lines $AP_i$, $BP_i$ and $CP_i$ meet the circumcircle $(ABC)$ again at $D_i$, $E_i$ and $F_i$, respectively, and the segments $D_iE_i$ and $D_iF_i$ satisfy $D_iE_i\perp D_iF_i$ and $D_iE_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 | 1dcc81e0-7ea3-5326-863d-e59c2086f2f5 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $k$ be the circumcircle of triangle $ABC$ with $AC<BC$. Let $L$ be the foot of the internal angle bisector of $\angle ACB$ on side $AB$ ($L\in AB$). Let $M$ be the midpoint of the arc $AB$ of $k$ that contains $C$. Let $I$ be the incenter of $\triangle ABC$. Let $K\neq M$ be the second intersection of line $MI$ wit... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 083bbd8d-353c-5495-b537-0ec15445d100 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Show that there exists a partition of the positive integers into two subsets $A$ and $B$ such that
(i) for any three consecutive positive integers, at least one of them belongs to $A$ and at least one belongs to $B$ (i.e., no three consecutive integers are all in the same subset), and
(ii) neither $A$ nor $B$ conta... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 2fa56305-b895-5016-a1ef-a477bd477f03 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ with $abc=1$. Prove that
\[
\frac{1}{a^{7}+b^{7}+c}+\frac{1}{b^{7}+c^{7}+a}+\frac{1}{c^{7}+a^{7}+b}\le 1.
\]
Equality holds when $a=b=c=1$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 3ce4d972-b2b5-5b52-940c-6ae8015130d7 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with circumcircle $(O)$ and orthocenter $H$. Let $P$ be any point on the minor arc $BC$ of $(O)$. Let $P'$ be the reflection of $P$ across line $BC$. Let $\omega$ be the circumcircle of $\triangle OPP'$ and let $G$ be the second intersection of $\omega$ with line $AP$ (i.e. $G\neq A$). 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 | 9b0805a1-7e93-5c56-8410-f6e188dd9550 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given an acute triangle $ABC$ with circumcircle $(ABC)$. Let $D$ be a fixed point on the minor arc $BC$ (the arc not containing $A$). Let $I$ be a variable point on the segment $AD$. Define
\[
E=BI\cap AC,\qquad F=CI\cap AB,
\]
let $M,N$ be the second intersections of the line $EF$ with $(ABC)$, and define
\[
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 | cc47fa17-b56d-5fcd-8380-7de9bb21b81a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In $\triangle ABC$, let $I$ be the incenter and $I_a$ the excenter opposite $A$. Let $A' = II_a\cap BC$. Let $M$ be the midpoint of the arc $BC$ of the circumcircle $(ABC)$ that does not contain $A$. Let $N$ be the midpoint of the arc $MA$ of $(ABC)$ that contains $B$ (i.e., the midpoint of the arc from $M$ to $A$ pass... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a172b0df-0b68-5f8d-9895-7ce18be3470b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(x,y,z\in\mathbb R\) be real numbers such that the denominators are non‑zero \((x^2+y^2\neq0,\;y^2+z^2\neq0,\;z^2+x^2\neq0)\). Prove that
\[
\frac{x(x+y)}{x^{2}+y^{2}}+\frac{y(y+z)}{y^{2}+z^{2}}+\frac{z(z+x)}{z^{2}+x^{2}}\le 3.
\] | {
"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 | 0a5d8699-998f-5db0-b2b0-2f5a7079c929 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a cyclic quadrilateral inscribed in a circle $(O)$ with vertices in that order. Let $E=AC\cap BD$. Let $\ell$ be any line through $E$ intersecting $AB$ at $P$ and $BC$ at $Q$. Let $\omega$ be the circle through $D$ that is tangent to $\ell$ at $E$. Let $R\neq D$ be the second intersection point of $\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 | bea3c9c8-44b4-5ca5-b796-e3bf5a3a93b4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Suppose that the function $f:\mathbb{R}\to\mathbb{R}$ satisfies
\[
f(x^{3}+y^{3})=(x+y)\bigl(f(x)^{2}-f(x)f(y)+f(y)^{2}\bigr)
\]
for all real numbers $x,y$. Prove that
\[
f(1996x)=1996\,f(x)\qquad\text{for every }x\in\mathbb{R}.
\] | {
"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 | 2d636744-bf32-509f-a1c9-c7354921518b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given the list of positive integers $1,2,3,4,\dots$, at each step take the three smallest numbers that have not yet been removed, record their sum, and then cross off those three numbers together with their sum. This produces a sequence of recorded sums: $6,16,27,36,\dots$. Prove or disprove that some term of this sequ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 39bddf85-c3a4-5544-9376-cc36fc7d1e86 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $P$ and $Q$ be two isogonal conjugate points with respect to triangle $ABC$. Let $K$ and $L$ be the orthogonal projections of $P$ and $Q$ onto line $BC$. Let $N$ and $M$ be the points on the pedal circles of $P$ and $Q$ such that $KN$ and $LM$ are diameters of those circles. Let line $MN$ intersect 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 | bed2c9a3-565f-5333-9dde-7a45cc42438c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a cyclic quadrilateral (its vertices lie on a circle $O$).
Define
\[
E=AB\cap CD,\qquad F=BC\cap AD,\qquad G=AC\cap BD.
\]
Let $H$ be the second intersection point of the circumcircles of $\triangle DCF$ and $\triangle DAE$ (the Miquel point of the complete quadrilateral formed by the lines $AB,BC,C... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 1c3086d7-56ba-5960-9514-ce86055b99db | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be positive real numbers. Prove that
\[
(ab+bc+ca)\Big(\frac1{a^2+b^2}+\frac1{b^2+c^2}+\frac1{c^2+a^2}\Big)\ge\frac52.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | f2c06f21-5b2b-5574-b3d7-37525991fe0c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all pairs of integers \((m,n)\) such that
\[m^{6}=n^{\,n+1}+n-1.\] | {
"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 | e80a9906-973c-5ec9-825d-9d520f359e61 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb Z\to\mathbb Z$ such that
\[f(m^2+n^2)=f(m)^2+f(n)^2\]
for all integers $m,n$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 7f30ef46-6e75-59bc-8ba2-75d4d6ab0138 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\ge0$ satisfy $ab+bc+ca=3$. Prove that
\[
\sum_{\text{cyc}}\frac{a\,(b^{2}+c^{2})}{a^{2}+bc}\ge 3,
\]
with equality when $a=b=c=1$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 6b641946-37a2-511c-b38e-b58bc1c3d531 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | a) Do there exist infinitely many finite nonconstant arithmetic sequences $a_1,a_2,\dots,a_n$, where $n>2$, such that $\overline{a_1a_2\ldots a_n}$ is a palindrome?
b) Does there exist a finite nonconstant geometric series $g_1,g_2,\dots,g_n$, where $n>2$, such that $\overline{g_1g_2\ldots g_n}$ is a palindrome?
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 | 0fcf90f6-9dd4-57c1-bc9b-99f3be4a0481 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $(a_n)_{n\ge 1}$ be a sequence of positive real numbers. Assume there exists a constant $M>0$ such that for every $n\ge1$
\[
a_1^{2}+a_2^{2}+\dots +a_n^{2}<M\,a_{\,n+1}^{2}.
\]
Prove that there is a constant $M'>0$, depending only on $M$, such that for every $n\ge1$
\[
a_1+a_2+\dots +a_n<M'\,a_{\,n+1}.
\] | {
"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 | a8f3dab1-d667-5c7f-ad16-dc0b04ba4719 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCDEF$ be a cyclic hexagon and let $P$ be any point in the plane. For each of the six triangles $PAB$, $PBC$, $PCD$, $PDE$, $PEF$, $PFA$, denote by $A',B',C',D',E',F'$ respectively the circumcenters of these triangles. Prove that the three lines $A'D'$, $B'E'$, and $C'F'$ are concurrent (i.e., they intersect at 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 | 7a770a57-2940-5c51-93d7-52560fb48eb6 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $f:\mathbb{R}\to\mathbb{R}$. Prove that if
\[f(xy+y+x)=f(xy)+f(x)+f(y)\qquad\text{for all real }x,y,\]
then
\[f(x+y)=f(x)+f(y)\qquad\text{for all real }x,y.\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 9e47364a-e71d-5947-aa79-970a336a58a3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In triangle $ABC$, extend side $AB$ beyond $B$ to a point $D$ and side $AC$ beyond $C$ to a point $E$ such that the line $DE$ is parallel to $BC$ and $DE$ is tangent to the $A$‑excircle (the excircle opposite $A$). Construct the circle $\Omega$ that passes through $D$ and $E$ and is tangent to the $A$‑excircle. 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 | 8595e210-b307-5ec3-b223-0eaef36a39ab | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle. Points $P\in AB$ and $Q\in AC$ are chosen so that $PQ\parallel BC$. Let $O= BQ\cap CP$. Let $A'$ be the reflection of $A$ across line $BC$. Let $w$ be the circumcircle of $\triangle APQ$. The line $A'O$ meets $w$ again at $S$ (with $S\neq A$). Prove that the circumcircle of $\triangle BSC$ 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 | b9946912-3463-56da-bcfc-c6d692fc6ea2 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ satisfy $ab+bc+ca+abc=4$. Prove that
a) $abc+a+b+c\ge 4$,
b) $a^2+b^2+c^2+(abc)^9\ge 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 | aa0ece53-197c-5827-8fe7-3c7a9442031f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | If $a,b,c$ are positive real numbers, prove that
\[
\frac{a^{2}}{a+b}+\frac{b^{2}}{b+c}+\frac{c^{2}}{c+a}\le \frac{3\,(a^{2}+b^{2}+c^{2})}{2\,(a+b+c)}\;.
\] | {
"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 | 6c7fee3c-3ecd-5ca3-a38d-d850a6d33206 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a_1,a_2,\dots,a_n>0$ be real numbers with $a_1a_2\cdots a_n=1$ (where $n\in\mathbb N$). Prove that
\[
\frac{1}{a_1}+\frac{1}{a_2}+\dots+\frac{1}{a_n}+\frac{4n}{\,n+a_1+a_2+\dots+a_n\,}\ge n+2.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | b9ae6158-f522-5b7f-beb5-abd46a649119 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $K\subset\mathbb{R}^2$ be a compact convex set and let $G$ denote its centroid (center of mass). Prove that $G$ is the midpoint of at least three line segments whose endpoints lie on the boundary $\partial K$ (i.e., at least three chords of $K$). Moreover, determine whether this result can be strengthened to guaran... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a36ab419-1bef-559e-8efe-1b57114eeb82 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $(a_n)$ be defined by $a_0=0$, $a_1=1$, and
$$a_n=2a_{n-1}+a_{n-3}\qquad\text{for all integers }n\ge3.$$
Prove that for every integer $k\ge0$,
$$2^k\mid a_n\iff 2^k\mid n.$$
Let $a,b\in\mathbb{R}$ and define another sequence (also denoted $a_n$) by
$$a_1=a,\qquad a_2=b,\qquad a_{n+2}=\frac{a_{n+1}+a_n}{2}\qq... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 40e428b4-dae6-5c7b-a95c-d1d558871164 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c,d$ be non‑negative real numbers satisfying $ab+ac+ad+bc+bd+cd=6$. Prove that
\[
\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}+\frac{1}{d^2+1}\ge 2.
\] | {
"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 | 709c92dd-d95b-54b9-aba7-1bdccd2d5a2e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all integers $x,y,z$ satisfying
$$2x^4+2x^2y^2+y^4 = z^2.$$ | {
"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 | 25d4ce69-01ba-57b0-acee-39da5687f44e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given $N$ lines in the plane, no two of which are parallel and no three of which are concurrent, prove that there exists a non‑self‑intersecting broken line (simple polygonal chain) $A_0A_1A_2\cdots A_N$ with $N$ segments $A_{i-1}A_i$ ($i=1,\dots,N$) such that each of the $N$ given lines contains exactly one of these 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 | 4a490303-6007-5bb0-8384-92258c09dbb8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Winnie and Lizzie are playing tag inside a circular playground (a closed disc of radius R). Both beasts move continuously with the same constant speed v and are considered point objects. Can Winnie guarantee to catch Lizzie in a finite amount of time, no matter how Lizzie chooses to move? | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 9d042d1e-c424-53af-9c04-eb2d73227bbb | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a_1<a_2<\dots<a_n$ be a strictly increasing sequence of real numbers. Prove that the equation
\[
\frac{a_1}{a_1-x}+\frac{a_2}{a_2-x}+\dots+\frac{a_n}{a_n-x}=2015
\]
has exactly $n$ real (distinct) solutions. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 37ee8361-7760-5a53-96b8-660b371471c4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n$ be a positive integer. For $k\in\{1,2,\dots,2n^{2}\}$ consider all colorings of a $2n\times2n$ board with exactly $k$ black cells (the remaining cells are white). Let $s(k)$ denote the smallest possible number of $2\times2$ sub‑squares that contain both a black and a white cell. Find all $k$ for which $s(k)=2n-... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a5aeebcb-8d3d-502a-91b0-f60972dcbf9d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a non‑equilateral triangle. Points $M$ and $N$ lie inside $ABC$ such that
\[
\angle BAM=\angle CAN,\qquad \angle ABM=\angle CBN.
\]
Assume that
\[
AM\cdot AN\cdot BC=BM\cdot BN\cdot CA=CM\cdot CN\cdot AB=k
\]
for some real number $k$. Prove that
(a) $3k=AB\cdot BC\cdot CA$;
(b) the midpoint of th... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a6eb9845-b298-5341-b73e-16a79226a251 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $P,Q\in\mathbb{R}[x]$ be relatively prime non‑constant polynomials. Prove that there are at most three real numbers $\lambda$ for which $P+\lambda Q$ is the square of a polynomial (i.e. $P+\lambda Q=R^2$ for some $R\in\mathbb{R}[x]$). | {
"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 | a4ca2047-f51f-53b5-8626-6a36fe6e3ecb | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In triangle $ABC$, let $h_a$ be the altitude from $A$, $m_b$ be the median from $B$, and $t_c$ be the length of the internal angle bisector from $C$. Prove that
$$h_a+m_b+t_c\le\frac{\sqrt3}{2}\,(a+b+c),$$
where $a,b,c$ are the side lengths opposite $A,B,C$, 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 | f35aa368-1e5e-5248-bc59-30f812326b6b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Circles $O_1$ and $O_2$ intersect at points $X$ and $Y$. Let $\ell$ be a line tangent to $O_1$ at $A$ and to $O_2$ at $B$, with $O_1$ and $O_2$ on the same side of $\ell$ and $X$ nearer to $\ell$ than $Y$. The line $AX$ meets $O_2$ again at $P$. The tangent to $O_2$ at $P$ meets $AB$ at $Q$. Prove that $\angle XYB = \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 | 6f0643e9-6b5f-5f54-a235-8a4a0c5aca04 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $f:\mathbb{R}\to\mathbb{R}$ satisfy
\[f(x+y+xy)=f(x)+f(y)+f(xy)\qquad\forall\,x,y\in\mathbb{R}.\]
Prove that $f$ also satisfies the Cauchy additive equation
\[f(x+y)=f(x)+f(y)\qquad\forall\,x,y\in\mathbb{R}.\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 185821e3-ab13-5c83-8439-485b89f1b1d4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with incircle $(I)$ and circumcircle $(O)$. For any point $M$ on $(O)$, draw the two tangents from $M$ to $(I)$; let $M_1$ and $M_2$ be their intersections with side $BC$. Prove that the circumcircle of $\triangle M M_1 M_2$ always passes through a fixed point on $(O)$, namely the point where th... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 441a62ff-3132-5651-82d9-bc9f56aeb93d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$. Prove that
\[
\sqrt{\frac{a^{2}\bigl(a^{2}+bc\bigr)}{b^{2}+c^{2}}}+
\sqrt{\frac{b^{2}\bigl(b^{2}+ca\bigr)}{c^{2}+a^{2}}}+
\sqrt{\frac{c^{2}\bigl(c^{2}+ab\bigr)}{a^{2}+b^{2}}}\ge a+b+c\,.
\] | {
"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 | 556b5a38-eb7e-5669-93b2-504c0d4f2c43 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute triangle and let $DEF$ be its orthic triangle (so $D,E,F$ are the feet of the altitudes from $A,B,C$ respectively).
Show that
a) The radical center of the three circles $\odot(A,AD),\;\odot(B,BE),\;\odot(C,CF)$ (each circle has its center at the indicated vertex and radius equal to the dista... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | fb09adc3-5978-5759-b0b7-f46872de434c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,y,z>0$ satisfy $xyz=1$. Prove that
\[
\sqrt{\frac{x^2y+x^2z}{(y+z+1)^2}}+\sqrt{\frac{y^2z+y^2x}{(z+x+1)^2}}+\sqrt{\frac{z^2x+z^2y}{(x+y+1)^2}}\ge\sqrt2.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 33ee6c1e-6177-50b1-8112-a0529e27fb5a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\omega$ be a circle with centre $O$. Let $T$ be a point exterior to $\omega$ and let the tangents from $T$ to $\omega$ meet $\omega$ at $B$ and $C$, respectively. Choose points $K\in TB$ and $H\in TC$.
Define $B'$ (resp. $C'$) as the second intersection of the line $OB$ (resp. $OC$) with $\omega$.
Let $K'$ be 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 | 980971de-5a27-5a8a-ab1e-811f0b6388a2 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle, and let $O$, $I$, $I_a$ denote its circumcenter, incenter, and $A$‑excenter, respectively. Let $l$ be the line through $I_a$ that is perpendicular to $OI$. Prove that the line $l$, the side $BC$, and the perpendicular bisector of $AI_a$ are concurrent (i.e., they meet at a single point). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 23793e3f-9b9e-5217-847b-947b07e2680f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb Q^{+}\to\mathbb Q^{+}$ such that for every $x\in\mathbb Q^{+}$,
\[
\begin{cases}
f(x)+f\!\left(\frac{1}{x}\right)=1,\\[4pt]
f(2x)=2\,f\!\bigl(f(x)\bigr).
\end{cases}
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 34a70d96-5f0c-5ed4-9336-1948819fa0b8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle and $(I)$ its incircle. Let $D,E,F$ be the points where $(I)$ touches $BC,CA,AB$, respectively. Choose two distinct points $E',F'$ on the minor arc $EF$ of $(I)$ and a point $D'$ on the major arc $EF$ such that the lines $AD',\,BE',\,CF'$ are concurrent. Define
\[
X = EF\cap E'F',\qquad
Y = F... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 05c3dc33-a3a2-58bc-840c-bc9e7bbac746 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | If $a,b,c$ are positive real numbers, prove that
\[
\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\ge 3\sqrt{\frac{a^{2}+b^{2}+c^{2}}{ab+bc+ca}}.
\] | {
"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 | 77d3818d-fdea-5a0e-a9f1-aca489eaef57 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all natural numbers $n, k, l, m$ with $l>1$ such that
\[
(1+n^{k})^{l}=1+n^{m}\, .
\] | {
"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 | 3bed764b-ff9a-50e6-a878-919971d3d6d0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
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