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 $a,b,c$ be the sides of a triangle. Prove that
\[
\frac{a}{b^{2}+ca}+\frac{b}{c^{2}+ab}+\frac{c}{a^{2}+bc}\ge \frac{(a+b+c)^{3}}{6\,(ab^{3}+bc^{3}+ca^{3})}\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 5f618ef2-2078-5f5b-bc93-82965ca47c90 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For non‑negative real numbers \(x,\;y,\;z\) prove that
\[
\sum_{cyc}\frac{(z+x)(z+y)}{x^{2}+xy+y^{2}}\ge 4,
\]
where the sum is taken over the cyclic permutations of \((x,y,z)\). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 57ad3748-33e7-5a65-9a34-7135f815e867 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that for all positive real numbers $a,\;b,\;c$,
\[
\sqrt{a(a+b)(a+c)}+\sqrt{b(b+c)(b+a)}+\sqrt{c(c+a)(c+b)}\;\ge\;\frac{2}{3}\,(\sqrt a+\sqrt b+\sqrt c)\,(a+b+c).
\]
Equality occurs when $a=b=c$. | {
"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 | 7dcbf332-0ab2-562f-8f6b-887f97038590 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute scalene triangle with incircle $\omega$, and let $D$ be the point where $\omega$ is tangent to $BC$. For any point $X$ on the segment $AD$, let $M\neq X$ be the second intersection of line $BX$ with $\omega$, and let $N\neq X$ be the second intersection of line $CX$ with $\omega$. Prove that the l... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | f30f43b3-057a-501f-a77d-3b37eb90ea93 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | The $n$ contestants of EGMO are named $C_1,C_2,\dots ,C_n$. After the competition they form a queue in front of a restaurant according to the following rules.
1. The Jury may choose any initial order of the $n$ contestants in the queue.
2. Each minute the Jury chooses an integer $i$ with $1\le i\le n$.
* If con... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 569016a8-560d-5604-820e-3a7a7c4dc500 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Does there exist a tournament with the following conditions?
1. There are two sides, A and B, each containing six teams: A₁,…,A₆ and B₁,…,B₆.
2. There are six distinct sports.
3. The tournament consists of six rounds. In each round each team plays exactly once, so there are six matches per round. Each match is between... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 0de61df2-3d0f-507e-bce3-8b6f895a0d88 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions \(f:[0,\infty)\to[0,\infty)\) such that for all non‑negative real numbers \(x,y\),
\[
f(x^2)+f(y)=f\bigl(x^2+y+x\,f(4y)\bigr).
\] | {
"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 | 93e60ccc-6066-5c97-a6aa-936b70d8f151 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function satisfying
$$f(x+f(x))=f(x)\qquad\text{for all }x\in\mathbb{R}.$$
Find all such functions $f$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 9da321a9-e303-5949-a494-52dfb712f8c9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | On the circumcircle $\omega$ of triangle $ABC$, let $D$ and $E$ be points on $\omega$. Let $AD$ intersect $BC$ at $X$ and $AE$ intersect $BC$ at $Y$. Let $D',E'$ be the reflections of $D,E$ across the perpendicular bisector of $BC$. Prove that the lines $D'Y$ and $E'X$ intersect again on $\omega$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 18568c71-f412-5979-8ae5-fa2190e7a41a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute triangle. Let $X$ be a point inside $ABC$ such that
\[
\angle BAX = 2\angle XBA\qquad\text{and}\qquad \angle XAC = 2\angle ACX.
\]
Denote by $M$ the midpoint of the arc $BC$ of the circumcircle $(ABC)$ that contains $A$. Prove that $XM = XA\,. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 381df183-3472-5359-8544-46468e5e0204 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a$, $b$, and $n$ be positive integers. A lemonade stand has $n$ cups, all initially empty. It has a filling machine and an emptying machine that operate as follows:
* If at any moment there are $a$ completely empty cups available, the filling machine spends the next $a$ minutes filling those $a$ cups simultaneous... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 56030ef9-25a0-5f09-a8f5-ebc334cb5560 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $(O)$ be a circle with centre $O$. Let $AB$ be a diameter of the circle and let $CD$ be a chord. The line through the centre $O$ meets the chord $AC$ at $R$ and the chord $BD$ at $S$ (so $R\in AC$, $S\in BD$). Prove that
\[
\angle RCS=\angle RDS\,.
\]
(Equivalently, the points $C,R,D,S$ 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 | 8ed04cdd-8525-5654-b7a6-ced306839b8d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n$ be a positive integer such that $p=n^{2}+1$ is prime. Find all integer pairs $(x,y)$ that satisfy
\[
x^{2}-(n^{2}+1)\,y^{2}=n^{2}\,.
\] | {
"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 | 617b854a-8f38-59c1-86f8-e21f1b739614 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For what positive integers $n$ does there exist an $n$th‑degree polynomial with rational coefficients that has $n$ distinct irrational roots which are consecutive terms of a geometric progression? | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a824c3c2-ff3a-523a-9c20-b0c5d1b5ff52 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let
\[
1+\frac12+\frac13+\dots+\frac1n=\frac{a_n}{b_n},
\]
where $\gcd(a_n,b_n)=1$. Prove that there exist infinitely many integers $n$ such that the denominator of the reduced harmonic sum strictly decreases, i.e. $b_{n+1}<b_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 | 41cea814-c745-53f8-9e27-788e66c95f5c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle. Let $D\in BC$, $E\in CA$, $F\in AB$ such that the cevians $AD$, $BE$, $CF$ are concurrent. Let $M\in EF$, $N\in FD$, $P\in DE$. Prove that the lines $AM$, $BN$, $CP$ are concurrent if and only if the lines $DM$, $EN$, $FP$ 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 | 37f38155-3521-5d90-9568-4882e7fe1bf4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle such that the internal angle bisector $l_a$, the median $m_b$, and the altitude $h_c$ are concurrent; likewise $l_b$, $m_c$, $h_a$ are concurrent and $l_c$, $m_a$, $h_b$ are concurrent. Prove that
\[
\frac{\sin C}{\cos B}+\frac{\sin A}{\cos C}+\frac{\sin B}{\cos A}\ge 3\sqrt3\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | f232a26d-6b0e-5322-9fe7-9b8743aef8b7 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Denote by $I$, $O$, $X$ the incenter, circumcenter and Fermat point of an acute triangle $ABC$. Let $J$ be the incenter of the pedal triangle of $X$ (the triangle formed by the feet of the perpendiculars from $X$ to the sides $BC$, $CA$, $AB$). Prove that
$$XJ\parallel OI.$$ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a63b4389-c992-5d9f-9053-f8fb09fa90a2 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Define a sequence of integers by $a_0=1$ and
$$a_n=\sum_{k=0}^{n-1}\binom{n}{k}a_k\qquad(n\ge1).$$
Let $m$ be a positive integer, $p$ a prime, and $q,r$ non‑negative integers. Prove that
$$a_{p^{m}q+r}\equiv a_{p^{\,m-1}q+r}\pmod{p^{m}}.$$ | {
"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 | 13a17275-24f1-5875-8948-5244e874b341 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find (prove the existence of) a convergent sequence $(z_n)_{n\ge1}$ in $\mathbb{R}^2$ such that the open disks with center $z_n$ and radius $1/n$ are pairwise disjoint. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 2a2d875f-1f40-5f8f-a270-b1d056bef0c7 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Fix an integer $k>2$. Two players, Ana and Banana, play a game as follows. An integer $n\ge k$ is written on a blackboard. The players move alternately, Ana moving first. On a move a player replaces the current number $m$ by an integer $m'$ satisfying $k\le m'<m$ and $\gcd(m,m')=1$. The player who cannot make a move lo... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | cb50b253-b089-5db4-becb-9024ef4bf030 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a right triangle with $AB\perp AC$.
Choose points $M_1,M_2\in AB$ and $N_1,N_2\in AC$ such that
* $A$ is the midpoint of $M_1M_2$ and also of $N_1N_2$, and
* $AB^{2}+AN_{1}^{2}=AC^{2}+AM_{1}^{2}\,. $
For $i,j\in\{1,2\}$ define $K_{ij}=BN_i\cap CM_j$.
Prove that the quadrilateral $K_{11}K_{12}K_{... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | dc4bdb1c-8c7e-5521-ada1-ab8035586162 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $w_1$ be a circle with minor arc $AB$ and let $M$ be the midpoint of that minor arc. A second circle $w_2$ lies entirely inside $w_1$ and is not concentric with $w_1$. The line $AB$ meets $w_2$ at two distinct points $X$ and $Y$. Let $C$ and $D$ be the points on $w_2$ where the tangents from $M$ touch $w_2$ (so $MC... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 6e3cf929-8669-592b-bbd3-7b561fb0a6da | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with circumcenter $O$ and incenter $I$. For each vertex $X\in\{A,B,C\}$, let the internal angle bisector at $X$ intersect the circumcircle $(ABC)$ again at $X_1$ (so $A_1$, $B_1$, $C_1$ are the second intersections of $AI$, $BI$, $CI$ with the circumcircle). Let the line through $O$ parallel to ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | b2c43c7d-0bfb-55d3-9517-4d3266987c78 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let two circles, a smaller one Γ and a larger one Ω, be such that there exists a quadrilateral ABCD which is bicentric: Γ is the incircle of ABCD and Ω is the circumcircle of ABCD (we call such a pair of circles “good”).
(i) Choose points A′, B′, D′ on Ω so that the segments A′B′ and A′D′ are tangent to Γ. Choose 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 | 954661ff-19d2-5929-b0b2-e6ad211ecc43 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine all functions \(f:[1,\infty)\to[1,\infty)\) such that
(i) \(f(x+1)=\dfrac{f(x)^2-1}{x}\) for all \(x\ge 1\);
(ii) the function \(g(x)=\dfrac{f(x)}{x}\) is bounded on \([1,\infty)\). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | cdf93b0f-7245-5faa-8056-140a80d2cbee | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | A chess tournament took place between $2n+1$ players. Every player played every other player exactly once, with no draws. Each player had a distinct pre‑tournament rating. Exactly $k$ games were upsets, i.e., the lower‑rated player defeated the higher‑rated player. Prove that there exists a player who won at least $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 | f910634f-c8e0-56aa-b433-91a12633c221 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | If a function $f:\mathbb R\to\mathbb R$ satisfies
$$f'(x)\le f'\!\Bigl(x+\frac1n\Bigr)$$
for every real $x$ and every natural number $n\in\mathbb N$, prove that $f$ is continuously differentiable; i.e., $f'$ is continuous on $\mathbb R$. | {
"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 | 528b7ef5-c8cc-5bc5-ab18-73f77b27af87 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle and let $T$ be its centroid (the intersection of the three medians).
Let $p$ be a line passing through $T$ which meets the line $BC$ at $P$, the line $CA$ at $Q$, and the line $AB$ at $R$, with $Q$ and $R$ lying on the same side of $T$ along $p$. Prove that
\[
\frac{1}{TR}+\frac{1}{TQ}=\fra... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 5d92c017-dc5d-5342-814d-c23dcf1029ec | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $O$ be the circumcenter of $\triangle ABC$, where $\angle ABC>90^\circ$, and let $M$ be the midpoint of $BC$.
Point $P$ lies inside $\triangle ABC$ such that $PB\perp PC$.
Points $D$ and $E$, distinct from $P$, lie on the line through $P$ perpendicular to $AP$ and satisfy $BD=BP$ and $CE=CP$.
If quadrilateral... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 8f428418-af41-5547-8a6c-9b3ba9c383d3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Suppose the sequence of nonnegative integers $a_1,a_2,\dots,a_{1997}$ satisfies
\[
a_i + a_j \le a_{i+j} \le a_i + a_j + 1
\]
for all $i,j\ge1$ with $i+j\le1997$. Prove that there exists a real number $x$ such that $a_n=\lfloor n x\rfloor$ for every $n=1,2,\dots,1997\,$ (where $\lfloor\cdot\rfloor$ denotes the grea... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | e5314047-6db0-5953-970d-8948e85a8d79 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:[0,\infty)\to[0,\infty)$ such that for every $x,y\ge0$,
$$f(x^2)+f(y)=f\bigl(x^2+y+x\,f(4y)\bigr).$$ | {
"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 | edccd670-d1bb-51a5-ae77-cc11e9eccc58 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine all functions \(f:\mathbb{R}\to\mathbb{R}\) such that
\[
f\bigl(f(x)f(y)\bigr)=xy-x-y
\]
holds for all real numbers \(x,y\). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 51e9b726-6a19-5f54-b721-b544ce251ba3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(\{a_n\}\) be an increasing sequence of positive integers such that \(a_1 \le 2003\) and \(|a_{n+1}-a_n|\le 2003\) for all \(n\). Prove that there exist distinct indices \(n\neq m\) with \(a_n\mid a_m\). | {
"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 | 89c2af91-ba72-58e4-91ec-92b96c76ecaa | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | A frog starts at the integer 1 on the number line. If the frog is at an integer $n$, let $m_n$ be the exponent such that $2^{m_n}$ is the largest power of 2 dividing $n$ (i.e. $m_n=v_2(n)$). From $n$ the frog may jump either to $n+1$ or to $n+2^{\,m_n+1}$. For integers $k\ge2$ and $i\ge0$, let $f(n)$ denote the mini... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | f60354c3-b009-5546-84fb-b10116bd74fd | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For positive real numbers $a,b,c$ with $a+b+c=3$, prove that
\[
\frac{a}{b^{3}+1}+\frac{b}{c^{3}+1}+\frac{c}{a^{3}+1}\ge \frac32\,.
\] | {
"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 | 5bfba906-75ff-5aed-b322-95b1330cf16a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $P$ and $Q$ be isogonal conjugates with respect to $\triangle ABC$.
Let $D,E,F$ be the circumcevian triangle of $P$ with respect to $\triangle ABC$ (i.e. $D=AP\cap (ABC)$ other than $A$, $E=BP\cap (ABC)$ other than $B$, $F=CP\cap (ABC)$ other than $C$).
Let $X,Y,Z$ be the reflections of $Q$ in the midpoints of ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 6a04d486-3544-504f-a3bd-50341622731c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $D$ be a point on side $BC$ of triangle $ABC$, distinct from $B$ and $C$. Let $I$, $I_1$, and $I_2$ be the incenters of $\triangle ABC$, $\triangle ABD$, and $\triangle ACD$, respectively. Let $E$ be the second intersection point (different from $A$) of the circumcircles of $\triangle AI_1I$ and $\triangle ADI_2$, ... | {
"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 | a508279c-a7fc-5186-ac52-2c780362b4b3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\triangle ABC$ have circumcircle $(O)$ and incircle $(I)$. Denote the intouch triangle by $\triangle DEF$ (where $D$, $E$, $F$ are the points of tangency of $(I)$ with $BC$, $CA$, $AB$ respectively). Let $X$ be the foot of the perpendicular from $D$ to $EF$. Let $H$ be the orthocenter of $\triangle AXI$. Show that... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ee416664-3672-5da8-95c7-67a92290fd1d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(f:\mathbb N\to\mathbb R\) satisfy \(f(1)=1,\;f(2)=2\) and for every \(n\in\mathbb N\),
\[
f(n+2)=f\bigl(n+2-f(n+1)\bigr)+f\bigl(n+1-f(n)\bigr).
\]
Prove that for all \(n\) one has \(0\le f(n+1)-f(n)\le1\).
Determine all \(n\) such that \(f(n)=1025\). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | d627c904-e355-54bd-aa3d-d0d637547572 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b$ be positive integers such that
\[
\frac{a+1}{b}+\frac{b+1}{a}
\]
is an integer. Prove that
\[
\frac{a+b}{\gcd(a,b)^2}
\]
is a Fibonacci number. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | d82a09d9-e624-51f1-8f40-9bedddaecd75 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let a lattice polygon be a polygon whose vertices lie on points of the integer grid $\mathbb Z^2$. Denote by $I$ the number of lattice points strictly interior to the polygon and by $B$ the number of lattice points on its boundary (including the vertices).
(a) Prove Pick’s Theorem: the Euclidean area $A$ of the polyg... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 327ca9c5-64dd-5caa-a46f-ded959275f3e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let an equilateral triangle of side length $2012$ be tiled by unit equilateral triangles (a triangular lattice).
For each of the three directions of the lattice, consider every line segment of positive length inside the large triangle that is parallel to that direction and passes through vertices of the tiling (the t... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a34f2028-a1db-52bd-9866-075bd7cc6f99 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $P$ be a given point inside quadrilateral $ABCD$. Points $Q_1$ and $Q_2$ are inside $ABCD$ and satisfy
\[
\angle Q_1BC=\angle ABP,\qquad\angle Q_1CB=\angle DCP,\qquad
\angle Q_2AD=\angle BAP,\qquad\angle Q_2DA=\angle CDP.
\]
Prove that
\[
\overline{Q_1Q_2}\parallel\overline{AB}\iff\overline{Q_1Q_2}\parallel... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 582b726b-b979-5bb3-b3ef-aada62550236 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\Gamma$ be a circle and let $A$ and $C$ be two distinct points on $\Gamma$. Let $O$ be any point on the perpendicular bisector of $\overline{AC}$. Let $\omega$ be the circle with centre $O$ that meets the segments $OA$ and $OC$ at points $A'$ and $C'$, respectively. The tangents to $\omega$ at $A'$ and $C'$ inters... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 3379f789-6943-5f4d-a4cd-e40fd36d22a0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | A tournament on $2k$ vertices contains no directed $7$‑cycles. Prove that its vertex set can be partitioned into two subsets each of size $k$ such that the subtournament induced by each subset contains no directed $3$‑cycles (i.e., each part is transitive). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | c1e562f2-5d9d-5d13-be3c-31be3f6892db | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be positive real numbers such that
$$a^2b+b^2c+c^2a+abc=4.$$
Prove that
$$a+b+c\ge ab+bc+ca.$$ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 13d24ac0-b75d-55bd-982f-00d3cd4e0682 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine whether there exist two polynomials \(P\) and \(Q\) with integer coefficients, each of degree at least \(2018\), such that
\[P(Q(x)) = 3\,Q(P(x)) + 1\]
for all real numbers \(x\). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | d448f4c3-f82a-5dea-b72d-b5cdfe7aff31 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For all positive real numbers $a,b,c$, prove that
\[
\frac{(a+2b+c)^3}{3a^3+2b^3+3c^3}+\frac{(a+b+2c)^3}{3a^3+3b^3+2c^3}+\frac{(2a+b+c)^3}{2a^3+3b^3+3c^3}\le 24.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 0fdbd4a7-e7f0-5676-8555-9eea9e12ed63 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that for any three **distinct** positive numbers $a,b,c$, the three numbers can be side lengths of a (non‑degenerate) triangle (i.e. each sum of two exceeds the third) if and only if $f(a),f(b),f(c)$ can be side lengths of a triangle. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | b673170a-9ae3-5552-ac77-769265020745 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that
\[
\frac{a}{a^{2}+b^{3}+c^{3}}+\frac{b}{b^{2}+a^{3}+c^{3}}+\frac{c}{c^{2}+a^{3}+b^{3}}\le \frac{1}{5abc}\, .
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | bfcf9a34-3c44-5b32-b7b6-9080d65d84f7 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be an isosceles trapezoid with $AD\parallel BC$ inscribed in a circle $w$. Let $K$ be a fixed point on $w$ such that the line $BC$ separates $A$ and $K$. For a variable point $E$, construct the parallelogram $BCEF$ with $BF\parallel CE\parallel AK$. Let $B'$ and $C'$ be the second intersections of the 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 | e51a52a9-9419-50a9-a18e-93f51c8e6a76 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine all sequences $(p_n)_{n\ge1}$ of prime numbers for which there exists an integer $k$ such that the recurrence relation
\[p_{n+2}=p_{n+1}+p_n+k\]
holds for every positive integer $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 | 141e0d6c-9632-56aa-8304-2a90dcd2cfeb | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In an acute triangle $\triangle ABC$, let $M$ be the midpoint of side $BC$. Let $P$ be a point inside $\triangle ABC$ such that $AP$ bisects $\angle BAC$. The line $MP$ meets the circumcircle of $\triangle ABP$ again at $D$ and meets the circumcircle of $\triangle ACP$ again at $E$. Prove that if $DE = MP$, then $BC = ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | f425587f-0e4f-5e48-9639-0e0c9d4b7e9d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that for an integer $n>1$ and real numbers $a_0,a_1,\dots,a_n,k$ satisfying $a_1=a_{n-1}=0$, the following inequality holds:
\[
|a_0|-|a_n|\le \sum_{i=0}^{\,n-2}\bigl|a_i - k a_{i+1} - a_{i+2}\bigr|.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 1ffde4ee-3fde-5286-8f5b-b11db3b9261a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb N_0\to\mathbb R$ such that
$$f(w^2+x^2+y^2+z^2)=f(w^2+x^2)+f(y^2+z^2)$$
for all non‑negative integers $w,x,y,z\,$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 22e4012a-007c-5762-9bce-ffb92b2f8c51 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with circumcircle $(O)$ and incircle $(I)$. Through $B$ draw the line $\ell_B$ perpendicular to $OI$, and through $C$ draw the line $\ell_C$ perpendicular to $OI$. Choose points $P\in\ell_B$ and $Q\in\ell_C$ such that $PB=PI$ and $QC=QI$. Prove that the line $PQ$ is tangent to the incircle $(I)$... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 8c27286b-1034-5595-a515-2258f553f62b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For nonnegative integers $p$, $q$, $r$, let
$$f(p,q,r) = (p!)^p\,(q!)^q\,(r!)^r.$$
Compute the smallest positive integer $n$ such that for any triples $(a,b,c)$ and $(x,y,z)$ of nonnegative integers satisfying $a+b+c = 2020$ and $x+y+z = n$, the integer $f(x,y,z)$ is divisible by $f(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 | 27413e96-c564-50d5-87bf-a5a906de4af9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For which positive rational numbers $r$ does there exist infinitely many positive rational numbers $x$ such that
\[
\{x^2\}+\{x\}=r\,,
\]
where $\{x\}=x-\lfloor x\rfloor$ denotes the fractional part of $x\,$? | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 862b9b21-832b-5bc1-b655-7208a625cca5 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given $\triangle ABC$ with circumcenter $O$ and orthocenter $H$.
Let the line $OH$ intersect $AB$ at $E$ and $AC$ at $F$.
Let $S$ be the circumcenter of $\triangle AEF$.
Let the circumcircle of $AEF$ intersect the circumcircle of $ABC$ again at $J\neq A$.
Let line $OH$ intersect the circumcircle of $\triangle J... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | e752f4ca-eb8c-5e63-9d9e-6e5dad9e7353 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In triangle $ABC$ let $I$ be the incenter. The internal bisector $AI$ meets the circumcircle of $ABC$ again at $S\neq A$. Let $J$ be the reflection of $I$ across side $BC$. The line $SJ$ meets the circumcircle again at $P\neq S$. Prove that $AI=PI\,. $ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a0236554-29e8-5599-a4b1-d50479cfa915 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In a cyclic quadrilateral $ABCD$, let $M$ be the intersection of the extensions of opposite sides $AB$ and $CD$. Let $N$ be a point on the circumcircle of $ABCD$ such that the tangent to the circumcircle at $N$ passes through $M$ and the tangent line $MN$ is parallel to the diagonal $AC$. Let $P=NB\cap AC$. Prove that ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 08ff2245-2bd9-5992-b919-65c14c6dc548 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | A school has 450 students. Each student has at least 100 friends among the other students, and among any 200 students there are always two that are friends. Prove that 302 of the students can be chosen and placed in 151 two‑seater kayaks so that each kayak contains a pair of friends. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 7aa0e1b6-1a0e-5405-bb5f-bd189a043bdc | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In triangle $ABC$ with $AB\neq AC$, let its incircle be tangent to $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The internal angle bisector of $\angle BAC$ meets the lines $DE$ and $DF$ at $X$ and $Y$, respectively. Let $S$ and $T$ be distinct points on side $BC$ such that $\angle XSY=\angle XTY=90^\circ$. ... | {
"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 | 4e24e4ca-76b6-5798-9417-110262a9acff | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that in every pre‑Hilbert space (a vector space equipped with an inner product) the following inequality holds for all vectors \(a,b,c,d\):
\[
\|a-b\|\,\|c-d\| \;+\; \|b-c\|\,\|a-d\| \;\ge\; \|a-c\|\,\|b-d\|,
\]
where \(\|\cdot\|\) denotes the norm induced by the inner product. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 5c9539a7-a933-52e5-995b-5d1f7a2c2ad7 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Drake is thinking of a positive integer $x$. He tells Misha the number of digits $x$ has in base $2$, and he tells Ivy the number of digits $x$ has in base $3$. For example, $x=11=1011_2=102_3$, so Misha is told “$x$ has $4$ digits in base $2$” and Ivy is told “$x$ has $3$ digits in base $3$”.
**a.** Drake alternates ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | b4093fe8-fe38-55b1-959f-0f85593ba1e2 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In triangle $ABC$ with $AB=AC$, let $N$ be a point inside the triangle and let $M$ be a point outside the triangle such that $A$ is the midpoint of $MN$. Let $P$ be the intersection of the line through $B$ perpendicular to $BM$ and the line through $C$ perpendicular to $CM$ (equivalently, $P$ is the orthocenter of $\tr... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | b6a8c74b-baec-5488-b136-05b6106b609d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCDE$ be a cyclic pentagon. For any triangle $XYZ$ denote by $O_{XYZ}$ its incenter. Prove that the intersection of the medians of triangle $O_{ABC}O_{ACD}O_{ADE}$ coincides with the intersection of the medians of triangle $O_{ABE}O_{BED}O_{BCD}$ (i.e. the two triangles have the same centroid). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 557a81d3-8c5c-5e2b-8e62-06a8d3f9c844 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a convex quadrilateral circumscribed about a circle (i.e. a tangential quadrilateral). Assume that each interior angle and each exterior angle of $ABCD$ is at least $60^\circ$ (equivalently, every interior angle lies in $[60^\circ,120^\circ]$). Prove that
\[
\frac13\bigl|AB^{3}-AD^{3}\bigr|\;\le\;\bigl... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 010529fb-e4f8-5970-9b1f-402ad2186f59 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with circumcenter $O$, orthocenter $H$, altitude $AD$, and median $AM$. The line through $H$ perpendicular to $AM$ meets $AM$ at $P$ and $BC$ at $Q$. Let $T$ be the second intersection of line $OA$ with the circle $(BOC)$, and let $R$ be the second intersection of line $OD$ with $(BOC)$. Let $S$... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a1596e93-5df0-58a4-8df3-0e30f0535c54 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all non‑negative integers $a,b,c,d$ such that $5^a+5^b=2^c+2^d$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | d1562507-3ad3-5a13-ae39-fecc986d4fc3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all polynomials $P(x)\in\mathbb Z[x]$ of odd degree $d$ such that for every positive integer $n$ there exist $n$ positive integers $x_1,x_2,\dots,x_n$ satisfying
\[
\frac12<\frac{P(x_i)}{P(x_j)}<2\qquad\text{and}\qquad\frac{P(x_i)}{P(x_j)}\text{ is a $d$‑th power of a rational number}
\]
for all $1\le i,j\le 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 | d80dbdf6-d027-571a-a4da-d2a85e861ac6 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given an integer $n\ge 3$, let $n$ distinct points on a circle be labeled $1,2,\dots ,n$. A chord $[ij]$ connects point $i$ to point $j$ ($i\neq j$). Determine the maximum possible number of chords such that every pair of chosen chords has a non‑empty intersection (they meet either at an endpoint or in the interior of ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 1c96f0b1-10a1-5e24-9e71-c4b8fea1c5be | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $A$ be a skew‑symmetric matrix (i.e., $A^T=-A$) with real entries.
(a) Prove that $\det A\ge 0$.
(b) If $A$ has integer entries, prove that $\det A$ is the square of an integer. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 8114e2ad-0d31-5c7a-8f96-24a0d0c7f345 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | If the polynomials $f(x)$ and $g(x)$ are written on a blackboard then we may also write the polynomials $f(x)\pm g(x)$, $f(x)g(x)$, $f(g(x))$, and $c\,f(x)$, where $c$ is an arbitrary real constant. Initially the polynomials $x^{3}-3x^{2}+5$ and $x^{2}-4x$ are on the blackboard. After a finite number of such steps, can... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 9946b9b1-0120-5d30-9158-df2c132bf79b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a circumscribed quadrilateral and let $P$ be the orthogonal projection of its incircle center onto diagonal $AC$. Prove that $\angle APB = \angle APD\,. $ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 1deb3953-0a3e-5661-82b5-439004699647 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\mathbb Z_{\ge0}$ be the set of non‑negative integers, and let $f:\mathbb Z_{\ge0}\times\mathbb Z_{\ge0}\to\mathbb Z_{\ge0}$ be a bijection satisfying
\[
\text{if }f(x_1,y_1)>f(x_2,y_2)\text{ then }f(x_1+1,y_1)>f(x_2+1,y_2)\text{ and }f(x_1,y_1+1)>f(x_2,y_2+1).
\]
For $0\le x,y<100$ let $N$ be the number of pa... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 273e16a3-b3ef-59f8-80e0-e35aff2987a3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with circumcircle $(O)$ and incenter $I$. Let $G$ be the second intersection of line $AI$ with $(O)$ (the point distinct from $A$). Let $K$ be the reflection of $I$ across line $BC$. Let $P$ be the second intersection of line $GK$ with $(O)$ (the point distinct from $G$). Prove that $IA = IP\,. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 2da2c6c5-d31b-5e46-9788-5fa7d8c30810 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In $\triangle ABC$, let $D$ be the foot of the altitude from $A$ onto $BC$, and let $P$ be any point on the altitude $AD$. Define $E=BP\cap AC$ and $F=CP\cap AB$. Prove that $\displaystyle\angle ADE=\angle ADF\,. $ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a08b294d-d08c-5c1d-8a5c-9005f5fdc41c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a quadrilateral inscribed in a circle $k$. Let $l$ be the line through the midpoints of its diagonals $AC$ and $BD$. Let $M$ and $N$ be the intersection points of $l$ with $k$. Let $P$ be the intersection point of lines $AD$ and $BC$. Prove that $\angle APM = \angle BPN$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 349b393b-3be3-5279-a683-cd66a2407c9f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given three non‑negative real numbers $a,b,c$ with $ab+bc+ca>0$, prove that
\[
\frac{a}{\sqrt{b^{2}-bc+c^{2}}}+\frac{b}{\sqrt{c^{2}-ca+a^{2}}}+\frac{c}{\sqrt{a^{2}-ab+b^{2}}}\ge 2+\frac{3abc}{a^{3}+b^{3}+c^{3}}\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ea06f23c-a853-5be6-9a87-c0a6d2eeb66f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with altitudes $AD,\;BE,\;CF$, where $D\in BC$, $E\in CA$, $F\in AB$. Let $O$ be the circumcenter of $\triangle ABC$. Points $P$ and $Q$ lie on $DE$ and $DF$, respectively, such that $FP\perp AC$ and $EQ\perp AB$.
a) Prove that the midpoint $K$ of $AO$ is the circumcenter of $\triangle DPQ$.
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 | 0437d08d-69ce-5048-8b3e-50c42bf1660e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let ω₁ and ω₂ be two circles externally tangent at T.
Let X be a point on ω₁ and let l₁ be the tangent to ω₁ at X.
Let l be a line intersecting ω₂ at A and B.
The line XT meets ω₂ again at S.
Let C be a point on the arc TS of ω₂ that does not contain A or B.
Let Y be a point on ω₁ such that the line YC is tan... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | c4a87035-7253-5e0b-b4ff-829ad2477125 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In triangle $ABC$, let $E$ be the point where the incircle touches side $AB$. Let $\xi$ be the perpendicular bisector of $AB$, and let $D$ be the reflection of $C$ across $\xi$. Let the circle tangent to lines $AC$, $BC$, and the circumcircle of $ABC$ touch $AC$, $BC$, and the circumcircle at $X$, $Y$, and $F$, respect... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 471e5b5b-f094-534e-b8d7-1e44842ef009 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let c > 2, and let a(1), a(2), … be a sequence of non‑negative real numbers satisfying
\[
a(m+n)\;\le\;2\,a(m)+2\,a(n)\qquad\text{for all integers }m,n\ge1,
\]
and
\[
a(2^{k})\;\le\;\frac{1}{(k+1)^{c}}\qquad\text{for all integers }k\ge0.
\]
Prove that the sequence \{a(n)\} is bounded; i.e. there exists a consta... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 21bf3409-0ebf-5afe-941d-58c4e715e7b1 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$, and assume $BC$ is not parallel to $AD$. Let $P=AC\cap BD$. Let $E$ be the intersection of the lines $AB$ and $CD$. Let $I$ be the incenter of $\triangle EBC$, and let the incircle of $\triangle EBC$ touch $BC$ at $T_1$. Let $J$ be the excenter of $\triangle E... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | cd0b8e89-a72b-5d6c-bbf4-77aaaaf34c30 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $A,B,C,D$ be four points on a circle (in this order). Let $K=AC\cap BD$. Denote by $I_1,I_2,I_3,I_4$ the incenters of triangles $ABK$, $BCK$, $CDK$, $DAK$, respectively. Let $M_1,M_2,M_3,M_4$ be the midpoints of the (minor) arcs $AB$, $BC$, $CD$, $DA$ of the circumcircle. Prove that the four lines $M_1I_1$, $M_2I_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 | 45c1be02-c0c3-5376-a2d7-f37e8fdc925c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with circumcircle $(O)$. Choose arbitrary points $M,N$ on side $BC$.
Let $(I)$ be the circle tangent to the lines $AM$ and $BM$, touching $BM$ at $E$, and internally tangent to $(O)$.
Let $(J)$ be the circle tangent to the lines $AN$ and $CN$, touching $CN$ at $F$, and internally tangent to ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | f2c844f8-94ef-58af-a0f2-2d6e71c65334 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | The circles \(k_1\) and \(k_2\) intersect each other at points \(X\) and \(Y\).
Each of \(k_1\) and \(k_2\) is internally tangent to a third circle \(k\): \(k_1\) is tangent to \(k\) at \(P\) and \(k_2\) is tangent to \(k\) at \(Q\).
The line \(XY\) meets \(k\) again at points \(A\) and \(B\).
The line \(PA\) mee... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | a93ca44f-2130-545b-9023-8b15182d03cd | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be positive integers and let $x,y,z$ be integers such that
(a) $ax^2+by^2+cz^2=abc+2xyz-1$;
(b) $ab+bc+ca\ge x^2+y^2+z^2$.
Prove that each of $a,b,c$ can be written as a sum of three squares of 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 | 79e754c6-61e2-5af1-a624-7db8048c7069 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a$ and $b$ be non‑negative integers and let $c$ be an integer such that $ab\ge c^{2}$. Prove that there exist a non‑negative integer $n$ and integers $x_{1},\dots,x_{n},\;y_{1},\dots,y_{n}$ satisfying
\[
\sum_{i=1}^{n}x_{i}^{2}=a,\qquad
\sum_{i=1}^{n}y_{i}^{2}=b,\qquad
\sum_{i=1}^{n}x_{i}y_{i}=c.
\] | {
"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 | 49920467-8880-5bf6-bc30-414dfbff918b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all quadruplets of positive integers $(m,n,k,l)$ such that
$$3^m = 2^k + 7^n$$
and
$$m^k = 1 + k + k^2 + k^3 + \dots + k^l.$$ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | e8ff4b65-f1eb-5a35-b31b-c44164712e6c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Consider the recurrence relation
\[
a_{n+2}a_{n}=a_{n+1}^{2}+2,\qquad a_{1}=a_{2}=1,
\]
and assume that all terms \(a_{n}\) are integers. Prove that every \(a_{n}\) is odd and that for each \(n\in\mathbb N\) the three numbers \(a_{n},a_{n+1},a_{n+2}\) are pairwise coprime, i.e.
\[
\gcd(a_{n},a_{n+1})=\gcd(a_{n},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 | 1e7e18c7-4feb-5836-ad5e-3e32fa35d5d4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle. Let $AD$ be the internal angle bisector of $\angle A$, with $D$ on $BC$. Let $M$ be a point on $CA$ such that $DM\parallel AB$. Let $E$ be the intersection of the internal bisector of $\angle ABM$ with $AD$. Let the circumcircle of $\triangle ABE$ meet $BM$ again at $F$. Define $I=AF\cap BE$ an... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ca1b5bc9-7908-52f7-9774-e333d6096666 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle. Let the internal bisector of $\angle A$ meet the circumcircle of $ABC$ again at $A_{1}$; define $B_{1}$ and $C_{1}$ similarly. Let the line $AA_{1}$ intersect the external angle bisectors at $B$ and at $C$ at a point $A_{0}$. Define $B_{0}$ and $C_{0}$ analogously. Prove that
\[
\operatorname{... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 1e306b66-e544-5df5-a7a4-8db7c947ea36 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $k$ and $n$ be integers with $0\le k\le n-2$.
Let $L$ be a set of $n$ lines in the plane such that no two are parallel and no three are concurrent.
Denote by $I$ the set of all intersection points of lines in $L$.
Let $O$ be a point in the plane that does not lie on any line of $L$.
A point $X\in I$ is colo... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | b4861751-1964-58c7-bfcd-d99de04c7831 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all integer quadruples \((x,y,z,a)\) satisfying
\[
2^{x}+5^{y}+7^{z}=13^{a},
\]
where the exponents are non‑negative integers (so each term is an integer). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 0b455ad5-d020-57cf-8d0f-d3c16acff7fb | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with side lengths $a,b,c$, medians $m_a,m_b,m_c$, circumradius $R$, and inradius $r$. Prove that
1. $\displaystyle \frac{m_b+m_c}{m_a+2m_c}+\frac{b+c}{a+2c}\ge\frac{4}{3}.$
2. $\displaystyle \frac{c+a}{b+2c}+\frac{m_a}{2m_b+m_c}\ge 1.$
3. $\displaystyle \frac12\,R\Bigl(1+\frac{b+c}{a}\B... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 48954627-d8f2-5aaf-a55e-c5b1a0e96ce7 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(a,b,c>0\) satisfy \(a+b+c=ab+bc+ca\). Prove that
\[
\frac{ab+bc+ca}{3}+2\;\ge\;\sqrt{ab}+\sqrt{bc}+\sqrt{ca}.
\]
(The proof must not use the \(uvw\) method.) | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ee721e57-a886-573c-918b-03beaca4188a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | With $a,b,c>0$ and $(a+b)(b+c)(c+a)=1$, prove that
\[
\frac{a}{(a+2b)^2(b+c)}+\frac{b}{(b+2c)^2(c+a)}+\frac{c}{(c+2a)^2(a+b)}\ge \frac{2}{3}.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 9835f0f7-7c8a-52d1-bab4-1cdb0b398385 | [] | AoPS | Nemotron-Math-Proofs-v3 |
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