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 $a$ and $b$ be real numbers with $a,b\ge 1$.
Define $a_{1}$ as the unique real number satisfying $a^{b}=a_{1}^{\,a_{1}}$ and define $b_{1}$ as the unique real number satisfying $b^{a}=b_{1}^{\,b_{1}}$.
For $n\ge1$ define $a_{n+1}$ and $b_{n+1}$ by
\[
a_{n}^{\,b_{n}}=a_{n+1}^{\,a_{n+1}},\qquad
b_{n}^{\,a_{n}}... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 8e9fd641-19c8-5558-b926-0ff4466d2886 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For $\theta,\phi\in[0,2\pi]$ and $\alpha,\beta\ge 1$, prove that
\[
\sqrt{1-\frac{\cos\theta}{\alpha}}\;\le\;\sqrt{1-\frac{\cos(\theta-\phi)}{\alpha\beta}}+\sqrt{1-\frac{\cos\phi}{\beta}}\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a8066357-ca6e-5553-be8b-bd14a1a75fcb | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
\[
f(2x+f(y)) = f(f(y)) + x\,f(2y) + f(2x)
\]
holds for every real numbers $x$ and $y$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 180f2098-42ca-58c2-b5c4-49e79c151e3b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $C(O,R)$ be a circle with diameter $PQ$ (the upper half of $C(O,R)$ is a semicircle).
Let $AB$ be a segment such that $B\in PQ$, $A$ lies on the semicircle, and $AB\perp PQ$ (so $AB$ is a half‑chord of the semicircle).
Let a circle $C(O',R')$ be tangent to the line $PQ$ at a point $C\in PQ$ and tangent internal... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | dbce0cff-e6a9-5505-99e5-714d78ba39d7 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In a city there are $n$ straight streets (modeled as lines in the plane) such that every two streets intersect and no three streets pass through the same point. At each intersection one of the two streets must be designated as the main street and the other as the side street. Prove that it is always possible to make 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 | 9154b7e4-9153-52ba-9798-3d1d52669257 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In $\triangle ABC$ let $H$ be its orthocenter and let $AD$ and $BE$ be arbitrary cevians. Let $\omega_1$ and $\omega_2$ be the circles with diameters $AD$ and $BE$, respectively. Let $F$ and $G$ be the second intersections of $HD$ with $\omega_1$ and $HE$ with $\omega_2$. Let $P_1$ and $P_2$ be the second intersections... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 0e9b709d-33e5-5f39-a118-034072b1b69a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
\[
f\bigl(f(x)-2y\bigr)=2x-3y+f\bigl(f(y)-x\bigr)\qquad\forall\,x,y\in\mathbb{R}.
\] | {
"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 | 8013ce58-962b-5eb8-94f5-cc435ff1225d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(a,b,c>0\) with \(abc=1\). Prove that
\[
\frac{a^{2}+b^{2}}{c^{2}+a+b}+\frac{b^{2}+c^{2}}{a^{2}+b+c}+\frac{c^{2}+a^{2}}{b^{2}+c+a}\ge 2.
\] | {
"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 | f1202a46-ede4-515d-8438-e63de4d191ae | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $G$ be an oriented graph with no directed cycles (a directed acyclic graph) and let $v,w$ be two vertices of $G$. A **click** is an operation that selects a vertex of $G$ whose incident edges all point outward (i.e. a source) and reverses the direction of every edge incident to that vertex. A **click‑sequence** is ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | d83464f6-9ce9-5bef-8917-41443d4bca62 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given non‑negative real numbers \(a,b,c\), prove that
\[
\left(\frac{a}{b+c}\right)^{3}+\left(\frac{b}{c+a}\right)^{3}+\left(\frac{c}{a+b}\right)^{3}
+\frac{abc}{(a+b)(b+c)(c+a)}
\;\ge\;\frac12\left(\frac{a^{2}+b^{2}+c^{2}}{ab+bc+ca}\right)^{2}.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ca41b2ec-0baf-53d9-bde7-67cd6a1ca7b3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In a $9\times9$ chessboard, 46 unit squares are colored blue. Prove that there exists a $2\times2$ sub‑board containing at least three blue squares (equivalently, the maximum number of blue squares one can color without creating such a $2\times2$ block is 45). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 900628f6-e51a-591c-98a5-04c33b1a05a4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n$ be a positive integer and let $a_1,a_2,\dots,a_{2n+1}$ be positive real numbers. For each $k=1,2,\dots,2n+1$ define
\[
b_k=\max_{0\le m\le n}\frac{1}{2m+1}\sum_{i=k-m}^{k+m}a_i,
\]
where the indices are taken modulo $2n+1$ (so the sum is over the $2m+1$ consecutive terms centered at $k$ on the cyclic sequen... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 734ff2b6-ca85-54fa-a38e-657f5d6a0687 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be positive real numbers such that $abc=1$. Prove that
\[
\frac{a^{2}}{\sqrt{1+a}}+\frac{b^{2}}{\sqrt{1+b}}+\frac{c^{2}}{\sqrt{1+c}}
\;\ge\;
\sqrt{\frac{3\,(a^{2}+b^{2}+c^{2})}{2}}\;.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 6eac2623-38a2-5c4d-b771-e2c3f858ff3e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f\colon\mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$,
\[
\bigl(f(x)+xy\bigr)\,f(x-3y)+\bigl(f(y)+xy\bigr)\,f(3x-y)=\bigl(f(x+y)\bigr)^2.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | f313b659-bf3a-5f8e-8216-8ca4b25c016f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $A,B\in\mathcal{M}_{2}(\mathbb{R})$ be real $2\times2$ matrices satisfying $A^{2}+B^{2}=AB$. Prove that $(AB-BA)^{2}=O_{2}$. | {
"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 | e90e40e0-0847-57b2-b21c-dfda52a5f976 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $p>5$ be a prime. Suppose there exists an integer $k$ such that $p\mid k^{2}+5$. Prove that there exist positive integers $m,n$ satisfying
$$p^{2}=m^{2}+5n^{2}\,. $$ | {
"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 | d44451e3-baa0-5ff4-9bfe-4268352c94a2 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a cyclic quadrilateral. Define $E=AB\cap CD$ and $F=AC\cap BD$. Let $H$ be the second intersection (different from $F$) of the circumcircles of $\triangle ADF$ and $\triangle BCF$. 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 | 31cfe3ca-b7b6-5318-b83b-ad21ace9f48d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Consider triangle $ABC$ and an arbitrary line $\ell$. Let $\ell\cap AB=C'$, $\ell\cap BC=A'$, $\ell\cap AC=B'$. Draw the line through $B'$ perpendicular to $AC$ and the line through $C'$ perpendicular to $AB$; denote their intersection by $X_A$. Let $B_A$ be the intersection of the line through $B'$ perpendicular to $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 | 20b920d8-eba4-52fd-9a4a-c34315a6f206 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\omega$ be a circle internally tangent to a circle $\Gamma$ at $T$. Let $A,B\in\omega$. Let $C=AT\cap\Gamma\neq T$. Let $X_1,X_2$ be the two intersection points of $\Gamma$ with the tangent line to $\omega$ at $B$. Define $Y_1=AB\cap CX_1$ and $Y_2=AB\cap CX_2$.
a) Prove that $\triangle CY_1Y_2$ is isosceles.
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 | e352ee83-bebc-53cb-a228-213a8f69b398 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ with $abc=1$. Prove that
\[
\frac{(a^4b^9+b^4c^9+c^4a^9)^4}{27a^9b^9c^9}\ge\frac1{b^4c^4}+\frac1{c^4a^4}+\frac1{a^4b^4}\,.
\] | {
"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 | b5c8f9d1-69ad-5465-bac6-c91844241927 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given positive real numbers $a,b,c$ such that $abc=1$, prove that
\[
\frac{1}{a^{5}+b^{5}+c^{2}}+\frac{1}{b^{5}+c^{5}+a^{2}}+\frac{1}{c^{5}+a^{5}+b^{2}}\le 1.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 8f3e4bd0-fce2-5bcd-b7e3-fa946c32f0a4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\ge0$ satisfy $ab+bc+ca=1$. Prove that
\[
\sum_{\text{cyc}}\frac{(a^{2}+1)^{2}}{b^{2}+c^{2}}\ge 8.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 94a04833-97ee-5fbb-8839-c3c9444bd079 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Acute triangle $ABC$ is inscribed in its circumcircle $\omega$. Let $H$ and $O$ be the orthocenter and circumcenter of $ABC$, respectively. Let $M$ and $N$ be the midpoints of $AB$ and $AC$, respectively. The rays $MH$ and $NH$ intersect $\omega$ again at $P$ and $Q$, respectively. Let $R$ be the intersection of lines ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | fc684556-67e6-5aed-9e40-6dfb336e0db0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ with $ab+bc+ca=1$. Prove that
\[
\frac{a}{bc+a^{2}}+\frac{b}{ac+b^{2}}+\frac{c}{ab+c^{2}}\;\le\;\frac32\,(a+b+c)\,.
\] | {
"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 | 5163a659-6ab8-5e4c-a1d1-8f3216307915 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a cyclic quadrilateral.
Define $P=AD\cap BC$, $Q=AC\cap BD$, and $R=PQ\cap AB$.
Let $M\neq R$ be the second intersection of the circumcircle of $\triangle CDR$ with the line $AB$.
Prove that $M$ is the midpoint of $AB$ (i.e. $AM=MB$). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 3488e7fb-7ba6-5516-bb8d-9eaeb2955a5f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ with $abc=1$. Prove that
\[
\frac{1}{a+\sqrt{3a+1}}+\frac{1}{b+\sqrt{3b+1}}+\frac{1}{c+\sqrt{3c+1}}\ge 1,
\]
and determine when equality occurs. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | aaa6750a-c98c-5b0b-aef5-23a1c2833f5a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Determine the maximum value of
\[
\frac{abc}{(a^{2}+b)(b^{2}+c)(c^{2}+a)}.
\] | {
"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 | 040c275c-4b7b-553a-9953-61f3424c948d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Consider 100 coins with positive real denominations not exceeding 1. Their total value is 50. Determine the smallest constant \(C>0\) such that for any such collection of 100 coins there exists a division into two piles of 50 coins each whose total values differ by at most \(C\). | {
"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 | 18fef1d7-baff-5a8a-8b81-51b37b780a59 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$,
\[
f(x^{2}+f(y))=f(f(x))+f(y^{2})+2f(xy).
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 546a98ca-4f11-5340-8301-75367eed852b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCDEF$ be a hexagon circumscribing a circle $(O)$. Assume that $O$ is also the circumcenter of triangle $ACE$. Prove that the circumcircles of triangles $OAD$, $OBE$, and $OCF$ are coaxial (i.e., they share a common radical axis). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | e9cecfaf-f676-5793-b1c0-fdc2fe7b15f9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a trapezoid with $AB\parallel CD$. Let $T$ be a point inside the trapezoid such that $\angle ATD=\angle CTB$. Let the line $AT$ intersect the circumcircle of $ACD$ again at $K\;(K\neq A)$ and the line $BT$ intersect the circumcircle of $BCD$ again at $L\;(L\neq B)$. Prove that $KL\parallel AB$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 362aa7b9-6438-5f0c-8c3a-128f70efdeb1 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a scalene triangle with circumcenter $O$ and incenter $I$. Let $X$ and $Y$ be the midpoints of the minor arcs $\widehat{AC}$ and $\widehat{AB}$ of the circumcircle of $ABC$, respectively. Let $E=BX\cap AC$ and $F=CY\cap AB$. Let $S\in AB$ and $T\in AC$ be the feet of the perpendiculars from $X$ to $AB$ 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 | e100d51d-d984-5d14-9fe2-f9c3a7af2a2d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that for all real numbers $x,y,z\in[1,2]$ the inequality
\[
(x+y+z)\Big(\frac1x+\frac1y+\frac1z\Big)\;\ge\;6\Big(\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}\Big)
\]
holds. Determine all triples $(x,y,z)$ in $[1,2]^3$ for which equality occurs. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 478aa97f-be6c-5ddd-8cb5-8171f05b6aa7 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\triangle ABC$ be a scalene triangle with circumcircle $(O)$ and incircle $(I)$. Let $X$, $Y$, $Z$ be the midpoints of the arcs $BC$, $CA$, $AB$ of $(O)$ (the arcs opposite $A$, $B$, $C$, respectively). Under the inversion in the incircle $(I)$, each circle with diameter $IX$, $IY$, $IZ$ is mapped to a line. Denot... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 3e8a5d4c-1a96-5e79-9756-3b3d09877cff | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,y,z\in\mathbb{R}^+$ with $xyz=1$. Prove that
\[
\frac{x^8+y^8-x^4y-xy^4}{\,x^8+y^8+z^4\,}
+\frac{y^8+z^8-y^4z-yz^4}{\,y^8+z^8+x^4\,}
+\frac{z^8+x^8-z^4x-zx^4}{\,z^8+x^8+y^4\,}\ge 0.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 7038b4db-ab9f-5832-abd0-c28c7281f129 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Show that the equation $a^4+2b^4=c^4$ has no solutions in positive integers. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | e2fcb0a2-4264-5fc4-a33d-3f6b34f5ca91 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a non‑isosceles triangle. Let $H$ and $O$ be its orthocenter and circumcenter. Let $B_0$ and $C_0$ be the feet of the altitudes from $B$ to $AC$ and from $C$ to $AB$, respectively. Let $(AHO)$ be the circumcircle of $\triangle AHO$. This circle meets the circumcircle $(ABC)$ again at $K_A$, meets line $AB$... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a28a0f78-616d-5e0a-be67-0b5789c91c83 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ such that for all $x,y\in\mathbb{R}^{+}$
$$f\!\left(xy\!\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{x+y}\right)\right)
= f\!\left(xy\!\left(\frac{1}{x}+\frac{1}{y}\right)\right)
+ f(x)\,f\!\left(\frac{y}{\,x+y\,}\right).$$
(Equivalently $f\big(x+y+\frac{xy}{x+y}\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 | 1e9e0d58-941c-57f2-a6c5-1fccb90587ec | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a cyclic quadrilateral whose opposite sides are not parallel. Let $X=AB\cap CD$ and $Y=AD\cap BC$ be the intersection points of the extensions of the opposite sides. The internal bisector of $\angle AXD$ meets $AD$ at $E$ and $BC$ at $F$. The internal bisector of $\angle AYB$ meets $AB$ at $G$ and $CD$ at... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | cf1b8edd-57ed-5b90-93b4-3053f059adff | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be real numbers such that $a\ge b\ge 1\ge c$ and $a^2+b^2+c^2=3$. Prove that
(a) $abc+2(b-c)^2\ge 1$,
(b) $abc+2(a-b)^2\le 1$,
(c) $abc+\frac12\,(a-c)^2\le 1\,. $ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 40a3b67f-eb9a-5d51-944f-1de3dd000b93 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $m$ and $n$ be non‑zero integers of the same parity. Suppose that $m^{2}-n^{2}+1$ divides $n^{2}-1$. Prove that $m^{2}-n^{2}+1$ is a perfect square. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | bc21a094-be33-5658-8a1b-e871db9d14cc | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\ge 0$ be real numbers such that $ab+bc+ca=1$. Prove that
\[
\frac{1+a^{2}b^{2}}{(a+b)^{2}}+\frac{1+b^{2}c^{2}}{(b+c)^{2}}+\frac{1+c^{2}a^{2}}{(c+a)^{2}}\ge\frac52.
\] | {
"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 | 69483268-b939-5fa5-84de-d550682ffd0d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute‑angled triangle. Let $AD$ and $BE$ be the altitudes, intersecting at the orthocenter $H$. Let $M$ be the midpoint of $AB$. Denote by $\omega_{1}$ the circumcircle of $\triangle DEM$ and by $\omega_{2}$ the circumcircle of $\triangle ABH$. The circles $\omega_{1}$ and $\omega_{2}$ intersect at two ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | bc158bfd-7cd3-5396-bd78-9e44df9eaae6 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that for all real numbers $x,\;y,\;z$,
\[
(xy+xz)(xy+yz)(xz+yz)\;\ge\;8\bigl(y^{2}+z^{2}-x^{2}\bigr)\bigl(z^{2}+x^{2}-y^{2}\bigr)\bigl(x^{2}+y^{2}-z^{2}\bigr)
+64\,(x-y)^{2}(y-z)^{2}(z-x)^{2}.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 7f729871-820b-574e-9af4-16daf45c1c7a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all real values of $\alpha$ for which there exists exactly one function $f:\mathbb{R}\to\mathbb{R}$ satisfying
\[ f\bigl(x^{2}+y+f(y)\bigr)=\bigl(f(x)\bigr)^{2}+\alpha\,y \]
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 | 777ddb5b-a37a-50a8-be63-95c13f2eb74d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | There are $n$ students standing in a circle, one behind the other. Their heights satisfy $h_1<h_2<\dots<h_n$. A move (switch) is allowed when a student of height $h_k$ stands directly behind a student whose height is $h_{k-2}$ or less; in that case the two students may swap places. Prove that no sequence of moves can 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 | fb38a058-5b77-592d-9eb2-47510861d8b4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $(A,+,\cdot)$ be a ring with $1+1\neq0$. Let $U(A)=\{a_1,a_2,\dots,a_n\}$ be the set of all units of $A$. Suppose there exists a bijection $\sigma:\{1,2,\dots,n\}\to\{1,2,\dots,n\}$ such that
$$a_{\sigma(1)}\cdot a_{\sigma(2)}\cdots a_{\sigma(n)}=1.$$
Prove that $4\mid n$. | {
"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 | 48059303-fe60-5d84-8311-bcedc9feff06 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine all pairs of natural numbers \((x,n)\) that satisfy the equation
\[
x^{3}+2x+1 = 2^{n}.
\] | {
"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 | fa2e8567-d863-5655-bb4e-16b9baafd532 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$.
Let $X$ be the intersection of the internal angle bisectors of $\angle A$ and $\angle B$, $Y$ the intersection of the internal bisectors of $\angle B$ and $\angle C$, $Z$ the intersection of the internal bisectors of $\angle C$ and $\angle D$, and $W$ the in... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a9d5403f-11dc-5f01-a51d-bed1e4f3fead | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a scalene triangle with circumcircle $\Omega$ and incenter $I$. Ray $AI$ meets $\overline{BC}$ at $D$ and meets $\Omega$ again at $M$; the circle with diameter $\overline{DM}$ cuts $\Omega$ again at $K$. Lines $MK$ and $BC$ meet at $S$, and $N$ is the midpoint of $\overline{IS}$. The circumcircles of $\tri... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 1f4d760a-5e12-572a-81fe-c14f4d5f6668 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Two‑chess is played like ordinary chess except that each player makes two consecutive moves on his turn. Prove that the white player has a non‑losing strategy (i.e., White can guarantee that he does not lose). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | d74605e5-de04-5db8-9bb1-d66cd1dc3d37 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Consider an \(n\times n\) board with a token placed in every \(1\times1\) cell (so there are \(n^{2}\) tokens). An operation may be performed as follows: if two tokens occupy the upper‑left and lower‑right corners of any axis‑aligned rectangle of cells, move the token from the upper‑left corner to the lower‑left corner... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 3fa6bc55-23bb-5de8-b238-d95eac6fc7c9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ with $a+b+c=1$. Prove that
\[
\sum_{cyc}\sqrt{\frac{a}{\,b+1\,}}\le \frac32\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 4351c1ee-f42a-5265-8c2d-51fa5562bb0e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | There are more than three airports. For any two airports $A$ and $B$, if there is a flight from $A$ to $B$, then there is also a flight from $B$ to $A$.
A **route** is a sequence of distinct airports $A_0,A_1,\dots ,A_n$ such that for each $i=0,\dots ,n-1$ there is a flight from $A_i$ to $A_{i+1}$; the length 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 | 56d1cfaf-df8f-5fa3-a0c4-78712fb3ed3a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute triangle with $AB>AC$. Let $\Gamma$ be its circumcircle, $H$ its orthocenter, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. Let $Q$ be the point on $\Gamma$ such that $\angle HQA=90^{\circ}$ and let $K$ be the point on $\Gamma$ such that $\angle HKQ=90^{\circ}$. Assum... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ed272841-d71a-59fa-a79a-47dc875eed79 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In a cyclic convex hexagon $ABCDEF$, let $G = AB\cap DC$ and $H = AF\cap DE$. Let $M$ be the circumcenter of $\triangle BCG$ and $N$ be the circumcenter of $\triangle EFH$. Prove that the three lines $BE$, $CF$, and $MN$ are concurrent. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | aa26f98d-acd6-5014-9728-e05f8907328b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Solve in natural numbers \(x,y,n\) the Diophantine equation
\[
x^{2}-1 = y^{\,n}.
\] | {
"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 | 044744b1-0f3e-5583-b02a-b89b8ea17b0a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(a,b,c\ge 0\). Prove that
\[
a\sqrt{a^{2}+bc}+b\sqrt{b^{2}+ca}+c\sqrt{c^{2}+ab}\ge (a+b+c)\sqrt{\frac13\Bigl(a^{2}+b^{2}+c^{2}+ab+bc+ca\Bigr)}.
\] | {
"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 | b5829ce6-4dd8-5544-978c-334122a0c010 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | A natural number $n$ is given. Prove that
$$\sum_{\substack{p\text{ prime}\\ n\le p\le n^2}}\frac{1}{p}<2,$$
where the sum is taken over all primes $p$ satisfying $n\le p\le n^2$. | {
"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 | c98328cf-94db-51db-a0af-5a84eff9c391 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | The circle $\Gamma$ with center $O$ passes through point $A$. Let $T$ be a point on the line $OA$ beyond $A$ (so $T$ lies outside $\Gamma$). Let $\Pi$ be a circle with center $Q$ that is tangent to $OA$ at $T$. The circles $\Gamma$ and $\Pi$ intersect at two points; denote one of them by $C$. The tangent to $\Gamma$ at... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 21bfbd8b-8eb3-5746-bf29-f97848424955 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For all positive real numbers $a,b,c$, prove that
\[
\sqrt{\frac{(5a+3c)(5b+3c)}{(a+b)^2}}+\sqrt{\frac{(5b+3a)(5c+3a)}{(b+c)^2}}+\sqrt{\frac{(5c+3b)(5a+3b)}{(c+a)^2}}\ge 12,
\]
i.e. equivalently
\[
\frac{\sqrt{(5a+3c)(5b+3c)}}{a+b}+\frac{\sqrt{(5b+3a)(5c+3a)}}{b+c}+\frac{\sqrt{(5c+3b)(5a+3b)}}{c+a}\ge 12.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 9b6cdfce-5b8c-5167-9e4d-90191e0aad1d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In a simple (non‑self‑intersecting) non‑convex hexagon $ABCDEF$ with no pair of opposite sides parallel, the interior angles satisfy $\angle A=3\angle D$, $\angle C=3\angle F$, and $\angle E=3\angle B$, and the side lengths satisfy $AB=DE$, $BC=EF$, $CD=FA$. Prove that the three diagonals $AD$, $BE$, and $CF$ are concu... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | cb5d576b-04bc-551a-b417-a413e733a692 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given an acute triangle $ABC$ with circumcenter $O$. The points $B$ and $C$ are fixed on the circumcircle $(O)$ while $A$ varies on $(O)$. Let $AD$ be a diameter of $(O)$ (so $D$ is the antipode of $A$ and $O$ is the midpoint of $AD$). Let $E=CD\cap AB$ and $F=BD\cap AC$. Reflect the line $BC$ across the internal bisec... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 6726b92f-a9f4-53d4-8dcc-8a1a30ddaa11 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $R$ be the set of all functions
$$f(x,y)=a^{2}x^{2}+(2ac-b^{2})xy+c^{2}y^{2}$$
where $a,b,c\in\mathbb Z$ satisfy $b^{2}-4ac=17$.
Define $S$ as the sum of all positive integers $n\le150$ for which there exist $f\in R$ and integers $x,y$ such that
$$n = f(x^{2},\,y^{2}).$$
Find the remainder when $S$ is divided... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 52747022-dd52-553f-95da-9325ffb23b2b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be real numbers such that $a+b+c=1$. Prove that
\[
\frac{a}{a^{2}+1}+\frac{b}{b^{2}+1}+\frac{c}{c^{2}+1}\le\frac{9}{10}\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | b3e602c2-ab72-5522-abc6-8bf631bfeb22 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a tetrahedron such that each of its four faces can be contained in a circle of radius $1$ (i.e. the minimal covering circle of each triangular face has radius $\le 1$). Prove that the tetrahedron can be placed inside a sphere of radius $\displaystyle\frac{3}{2\sqrt2}\,$. | {
"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 | b0304ea6-8de3-5864-aa56-24f75a4fdd82 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a>1$ be a positive integer. Prove that there exists a non‑negative integer $n$ such that the number $2^{2^n}+a$ is not prime. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 3af86520-974a-5686-8755-3e5f4c3a3b96 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | There are 100 girls and 100 colors. For each color there are exactly 100 balls, so there are 10 000 balls in total. Initially each girl possesses exactly 100 balls (the initial distribution may be arbitrary, subject only to the total of 100 balls per color). A move consists of two girls swapping one ball each (each giv... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 0cd9bd8e-a3b2-5300-b9b1-25ef154768a8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute‑angled triangle with $AB\neq AC$. The circle with diameter $BC$ meets $AB$ and $AC$ at $M$ and $N$, respectively. Let $O$ be the midpoint of $BC$. The internal bisectors of $\angle BAC$ and $\angle MON$ intersect at $R$. Prove that the circumcircles of $\triangle BMR$ and $\triangle CNR$ have a se... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 0f5c94e1-6ac0-51c3-b165-a8dd7166e5b8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that for all real numbers $x,\;y,\;z$ the inequality
\[
\frac{x\,(y+z)^{3}}{(y+z)^{2}+1}\;+\;\frac{y\,(x+z)^{3}}{(x+z)^{2}+1}\;+\;\frac{z\,(x+y)^{3}}{(x+y)^{2}+1}
\;\le\;
\frac{(x+y+z)^{4}}{(x+y+z)^{2}+1}\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 1ace2e2c-6391-5c3e-bf0f-3b43ad2add25 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with $AC = 2AB$. Let the tangents to the circumcircle of $ABC$ at $A$ and $C$ intersect at $P$. Let $D\neq B$ be the second intersection of line $PB$ with the circumcircle. Prove that $DB = DC$, i.e. $BP$ bisects the (major) arc $BAC$ of 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 | e7f617f4-fd7a-5914-a76b-28166051a7d1 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,y,z>0$. Prove that
\[
\frac{x^{2}+y^{2}+z^{2}}{xy+yz+zx}-1\;\le\;\frac{2^{2/3}}{3}\Biggl(\frac{(y^{2}+z^{2})(z^{2}+x^{2})(x^{2}+y^{2})}{x(y+z)\,y(z+x)\,z(x+y)}-1\Biggr),
\]
and determine all triples $(x,y,z)$ for which equality holds. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 6662624d-d921-5194-bbbd-d75ffe5e68b9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $P\in\mathbb Z[x]$ be a nonconstant polynomial that has no integer zeros. Prove that there exists a positive integer $m\le 3\deg P$ such that $P(m)$ does not divide $P(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 | e205d98f-d11d-5f43-b761-b9e0583635b9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$ be real numbers such that $a^{2}+b^{2}+c^{2}=1$. Prove that
\[
\frac{a}{1+bc}+\frac{b}{1+ac}+\frac{c}{1+ab}\le \sqrt{2}\,.
\] | {
"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 | 00636aa7-4b2f-52ba-8da3-8973991f2ead | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find the largest constant \(k\) such that for all positive real numbers \(a,b,c\) the inequality
\[
\frac{a}{b}+\frac{b}{c}+\frac{c}{a}-3\;\ge\;k\Bigg(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}-\frac32\Bigg)
\]
holds. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 80b63fc7-9347-5d60-b6b0-8e376ffdf2c2 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x_1,x_2,x_3,x_4\ge0$. Prove that
\[
\Bigg(\frac{x_1+x_2+x_3+x_4+x_1x_2+x_2x_3+x_3x_4+x_4x_1+x_1x_3+x_2x_4}{10}\Bigg)^2
\;\ge\;
\frac{x_1x_2x_3+x_2x_3x_4+x_3x_4x_1+x_4x_1x_2+x_1x_2x_3x_4}{5}.
\]
(Equivalently, with $S_k$ the $k$‑th elementary symmetric sum of $x_1,\dots,x_4$,
$(S_1+S_2)^2\ge20\,(S_3+S_4)$.) | {
"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 | 6795d549-0589-5bdd-8cab-94b9ff812bc7 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\ge0$ with $a+b+c=3$. Prove that
\[
\sum_{\text{cyc}}\frac{ab}{3+bc}\le\frac34.
\]
Equality occurs for $(a,b,c)=(1,1,1)$ and $(a,b,c)=(\tfrac32,\tfrac32,0)$ (and permutations). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 5de89f66-c448-5771-a39e-7d1da2113460 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | A line $d$ in the plane of a triangle $ABC$ is called **faithful** to $ABC$ if the reflections of $d$ across the three sides $AB,\;BC,\;CA$ are concurrent. Prove that for any two acute triangles lying in the same plane, either there exists exactly one line that is faithful to both triangles, or there exist infinitely 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 | c70fcddb-1804-5a22-9d4f-c62c391c78bc | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with circumcenter $O$. Let $B'$ be the reflection of $B$ across line $AC$, and $C'$ be the reflection of $C$ across line $AB$. Let $T$ be the circumcenter of triangle $OBC$. Prove that the line $AT$ is perpendicular to the line $B'C'$. | {
"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 | e358cbf9-4c42-5f30-a4ef-5e600a7967f0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying
\[
f\bigl(f(x)^2-f(x)y+y^2\bigr)=x^2-xf(y)+f(y)^2\qquad\forall x,y\in\mathbb{R}.
\] | {
"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 | d728c635-142b-5447-9e2d-d4ef2ccaf664 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given $a,b,c>0$, prove that
\[
\sqrt{\frac{a^{2}+b^{2}+c^{2}}{ab+bc+ca}}\;+\;\sqrt{2\Bigl(\frac{ab}{a^{2}+ab+bc}+\frac{bc}{b^{2}+bc+ca}+\frac{ca}{c^{2}+ca+ab}\Bigr)}\;\ge\;1+\sqrt{2}\,.
\] | {
"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 | a19822e2-38e3-50b7-8e42-19896ff33bec | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a non‑isosceles triangle. Denote by $O$, $I$, $H$ respectively its circumcenter, incenter and orthocenter. Prove that $\angle OIH$ is obtuse. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | e2caee36-af9f-5e76-9504-4a6cf5f06a6c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle. Points $E,G$ lie on $CA$ and points $F,H$ lie on $AB$ such that quadrilaterals $BCEF$ and $BCGH$ are cyclic. Let $(BEG)$ and $(CFH)$ denote the circles through $B,E,G$ and $C,F,H$, respectively, and let $P$ and $Q$ be their two intersection points. Prove that $\displaystyle\angle PAB=\angle QAC... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 742bcf8c-1241-50a2-99b1-98574d7760b8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute triangle. Denote by $I_B$ and $I_C$ the $B$‑excenter and $C$‑excenter of $ABC$, and by $O$ its circumcenter. Let $E$ be the foot of the perpendicular from $B$ to $AC$, and let $Y$ be the point on $AC$ such that $\angle ABY=\angle CBY$ (i.e. $BY$ is the internal angle bisector of $\angle ABC$). Sim... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | d65a95d2-b107-5f78-b65f-1af8f0d2edde | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given a triangle $ABC$ with incenter $I$. Let $\omega$ be the $A$‑mixtilinear incircle of $\triangle ABC$, i.e. the circle tangent to side $CA$ at $P$, to side $AB$ at $Q$, and to the circumcircle of $ABC$ at $R$. Define $D = AR\cap PQ$ and $E = IR\cap BC$. Prove that $DE\parallel AI$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 5a4f744a-99d6-5c2e-acb9-6d5a666584ab | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $O$ and $H$ be the circumcenter and orthocenter of $\triangle ABC$, respectively. Define
$$D=AH\cap BC,\qquad E=BH\cap AC,\qquad F=CH\cap AB,$$
$$M=AB\cap ED,\qquad N=AC\cap FD.$$
Prove that
1. $OB\perp DF$ and $OC\perp DE;$
2. $OH\perp MN.$ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 9f08cc0f-674d-5e59-b20b-3db72520f442 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,y,z>0$ such that $x^2+y^2+z^2=3$. Prove that
$$xy^2+yz^2+zx^2\le 2+xyz.$$ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 98686452-07c6-54c7-8144-72b3633c0d73 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Triangle $ABC$ is inscribed in the circle $\omega$. The circle $\omega_{1}$ is tangent to $\omega$ internally and tangent to sides $AB$ and $AC$ at points $M$ and $N$, respectively. The circle $\omega_{2}$ is also tangent to $\omega$ internally and tangent to sides $AB$ and $BC$ at points $P$ and $K$, respectively. Pro... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 4a3a4c37-a113-5b8e-a413-25c05b84dcc2 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$ with $x\neq0$ and $y\neq0$,
\[
f\bigl(f(x)-y\bigr)+f\bigl(f(x)+y\bigr)=2x.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ec05d029-e956-55b0-8a14-4a9a23d93c07 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | There are three circles $w_1,w_2,w_3$. For each $i$ (indices taken modulo 3), let the circles $w_{i+1}$ and $w_{i+2}$ intersect at points $A_i$ and $B_i$, where $A_i$ lies inside $w_i$. Let $\gamma$ be the circle that lies inside $w_1,w_2,w_3$ and is tangent to each of them; denote by $T_i$ the point of tangency of $\g... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | f1b0ea7e-f964-58c8-8994-3f1c5378cb45 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | $a,b,c>0$ with $ab+bc+ca=3abc$. Prove that
\[
\frac{a^2}{a^2+b^3+c^3}+\frac{b^2}{b^2+c^3+a^3}+\frac{c^2}{c^2+a^3+b^3}\le 1.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 97107912-1a02-5f02-86f0-1458bbcde998 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all metric spaces $(X,\delta)$ such that every triangle formed by three points in $X$ is degenerate; that is, for any distinct points $a,b,c\in X$ one of the three distances equals the sum of the other two:
\[
\delta(a,b)+\delta(b,c)=\delta(a,c)\quad\text{or}\quad
\delta(a,b)+\delta(a,c)=\delta(b,c)\quad\text{or}\... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 516e79f0-8097-53b9-b49f-68338e61971d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all prime numbers \(p,q,r\) such that
\[
\frac{p^{2q}+q^{2p}}{\,p^{3}-pq+q^{3}\,}=r.
\] | {
"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 | f8370628-2000-5cbd-9528-4f09520ffc42 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\Omega$ be the circumcircle of quadrilateral $ABCD$, and let $E$ be the intersection of the diagonals $AC$ and $BD$. A line through $E$ meets $AB$ at $P$ and $BC$ at $Q$. A circle passing through $D$ is tangent to line $PQ$ at $E$ and meets $\Omega$ again at $R\neq D$. Prove that $B,P,Q,R$ 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 | 01327db1-0014-5ddf-81e0-0a800f9ca7ae | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For an odd positive integer $n$, consider an $n\times n$ chessboard whose unit squares are colored alternately black and white in the usual way, but with the four corner squares colored black. An L‑tromino is a shape formed by three unit squares connected in an L‑shape. Determine for which odd $n$ it is possible to cov... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 4ed6a02b-a119-5888-86dc-307740213216 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given any rectangle drawn in the plane (its side lengths are known), does there exist a finite ruler‑and‑compass construction that partitions the rectangle into a finite number of polygonal pieces which can be reassembled to form a square? | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 320ff087-3bdf-5270-97c4-fbf97b4c6c87 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For $a,b,c>0$ prove that
\[
\frac{2a}{a+b}+\frac{a+b}{a+c}+\frac{6a+2c}{3b+c}+\frac{4a+3b+c}{b+c}\ge\frac{32a}{2a+b+c}\,.
\] | {
"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 | 5884b20e-df6d-5b8e-bf66-665f6b45ce59 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $b,c\in\mathbb{Q}$ with $c\neq0$ and $b\notin\{-1,0,1\}$. Determine all functions $f:\mathbb{Q}\to\mathbb{Q}$ such that
\[
f\big(f(x)+y\big)+f(x)=f(y)+f(bx)+c\qquad\text{for all }x,y\in\mathbb{Q}.
\] | {
"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 | 4b3762f9-eaca-54a1-b1c0-03fd6efa90df | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a cyclic quadrilateral whose opposite sides are not parallel. Suppose points $P,Q,R,S$ lie in the interiors of the segments $AB,BC,CD,DA$, respectively, such that
\[
\angle PDA=\angle PCB,\qquad
\angle QAB=\angle QDC,\qquad
\angle RBC=\angle RAD,\qquad
\angle SCD=\angle SBA.
\]
Let $AQ$ intersect $BS... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | e336237b-91a6-5046-8ba4-fa12700ffd07 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine all prime numbers $p$ for which there exists a positive integer $k$ such that the base‑$p$ representation of $4k^{2}$ consists only of the digit $1$ (i.e. $4k^{2}=1+p+\dots+p^{d-1}$ for some integer $d\ge1$). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 7755dd5f-1ed8-5075-8c53-b324a3bdd06b | [] | AoPS | Nemotron-Math-Proofs-v3 |
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