Datasets:

nvidia
/

Nemotron-Math-Proofs-v3-RL

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

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Nemotron-Math-Proofs-v3-RL

Dataset Description:

Nemotron-Math-Proofs-v3-RL is a long-form mathematical reasoning dataset for reinforcement learning. The release contains 9,597 proof-generation prompts.

The dataset uses NeMo Gym-compatible, single-turn user prompts derived from hard proof problems in the AoPS subset of nvidia/Nemotron-Math-Proofs-v1. The train split asks the policy to produce a rigorous solution and self-evaluation. Policy responses and realized rewards are generated during training and are not stored in the file.

Full details about dataset construction can be found in our technical report An Open Recipe for IMO Gold: Training Nemotron for Olympiad Mathematics.

This dataset is ready for commercial or non-commercial uses.

Dataset Owner(s):

NVIDIA Corporation

Dataset Creation Date:

Created on: 07/01/2026

Versioning:

Nemotron-Math-Proofs-v3-RL

Previous Version(s):

nvidia/Nemotron-Math-Proofs-v1 : Previous proof dataset version used as the source of problems for this release.
nvidia/Nemotron-Math-Proofs-v2 : Previous release version.
nvidia/Nemotron-Cascade-2-SFT-Data : Contains a natural language proof subset from the same source problems.

Relationship to Previous Version(s): This release adds reinforcement-learning data for proof generation, based on the same family of hard proof problems used for Nemotron-Math-Proofs-v1.

License/Terms of Use:

This dataset is governed by the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Intended Usage:

This dataset is intended for:

Reinforcement learning for structured mathematical reasoning and proof generation using online verifier feedback.
Training LLMs to produce proof self-evaluations and identify gaps in mathematical arguments.
Research on proof generation and reinforcement learning with verifiable rewards.

Dataset Characterization

Dataset Composition and Generation

Problem Sourcing

Problems are derived from nvidia/Nemotron-Math-Proofs-v1, which collected proof-based problems from the AoPS community. The released records identify their source as AoPS.

Reinforcement-Learning Data Preparation

The file is formatted for NeMo Gym and routes examples through agent_ref:

Proof generation: Each raw problem is inserted into a prover prompt and routed to proof_simple_agent. The expected policy response contains a solution, a self-evaluation, and a boxed score. During training, a proof-verification judge can score the generated proof and optionally assess the self-evaluation.

Dataset Fields

The train split contains the following fields:

agent_ref: NeMo Gym agent routing information.
responses_create_params: A single user message containing the formatted policy prompt.
problem: The natural-language mathematical problem.
uuid, license, source, and dataset: Record identifier and release provenance fields. All released records use cc-by-4.0, AoPS, and Nemotron-Math-Proofs-v3, respectively.
metadata: An empty list in the released file.

Data Collection Method

Hybrid: Manually-Collected, Synthetic, Automated

Labeling Method

Hybrid: Manually-Labeled, Synthetic, Automated

Dataset Format

Modality: Text
Format: JSONL
Structure: A single train split containing proof-generation prompts for reinforcement learning.

Dataset Quantification

Split Samples:

Split	File	Samples	Unique Problems	Disk Size
`train`	`train.jsonl`	9,597	9,597	50,430,007 bytes

The reinforcement-learning file does not contain generated-token counts because it stores policy inputs rather than completed policy rollouts.

Reference(s):

Ethical Considerations:

NVIDIA believes Trustworthy AI is a shared responsibility and we have established policies and practices to enable development for a wide array of AI applications. Developers should work with their internal developer teams to ensure this dataset meets requirements for the relevant industry and use case and addresses unforeseen product misuse. Please report quality, risk, security vulnerabilities or NVIDIA AI Concerns here.