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 $ABCD$ be a convex quadrilateral. Let $\omega_A$ and $\omega_B$ be the incircles of $\triangle ACD$ and $\triangle BCD$, with centers $I$ and $J$, respectively. Besides the common external tangent $CD$, there is a second common external tangent to $\omega_A$ and $\omega_B$; let it touch $\omega_A$ at $K$ and $\omeg... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 423bcd43-a5dc-5cd2-bda7-0f55860f28a0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a cyclic quadrilateral with circumcircle $(O)$. The tangents to $(O)$ at $B$ and $C$ intersect the line $AD$ at points $M$ and $N$, respectively. Define $E = BN\cap CM$ and $F = AE\cap BC$. Let $L$ be the midpoint of $BC$. Prove that the circumcircle of $\triangle DLF$ is tangent to $(O)$ at $D$. | {
"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 | c6d965d4-0af3-50f9-8f13-5c4a25055ca7 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all integer solutions \((x,y)\) of the equation \(2x^2 - y^{14} = 1\). | {
"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 | 979eba74-b625-58a1-90f6-aa14b96402a4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle inscribed in a circle $(O)$ and let $I$ be its incenter. Let $(K)$ be the circle tangent to the sides $CA$ and $AB$ of $\triangle ABC$ and internally tangent to $(O)$. Let $M,N$ be points on line $AO$ such that $\angle ABM=\angle ICB$ and $\angle ACN=\angle IBC$. Let $L$ be the circumcenter 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 | e223429e-fc81-535c-96c3-d1d15149623c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a convex quadrilateral that has an incircle (i.e., it is circumscribed) and satisfies
\[
\angle ABC+\angle ADC<180^\circ,\qquad
\angle ABD+\angle ACB=\angle ACD+\angle ADB.
\]
Prove that one of the diagonals of $ABCD$ passes through the midpoint of the other diagonal (i.e. either $AC$ bisects $BD$ or $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 | 71b13133-38d0-5444-800e-3ffdad5c8969 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute triangle with orthocenter $H$. Let $D$ and $E$ be points on side $BC$. The circumcircle of $\triangle BHC$ intersects the lines $AD$ and $AE$ again at points $P$ and $Q$, respectively. Prove that if
$$BD^{2}+CD^{2}=2\,DP\cdot DA\qquad\text{and}\qquad BE^{2}+CE^{2}=2\,EQ\cdot EA,$$
then $BP=CQ\,.... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 57321078-a09b-5da5-87ac-cd344ecfff08 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with circumcircle $(O)$ and incircle $(I)$. Let $\ell$ be any line through vertex $A$ intersecting $(O)$ again at $K$. Choose any point $X\in\ell$ such that $X$ lies outside the incircle $(I)$. The tangents from $X$ to $(I)$ meet the side $BC$ at $Y$ and $Z$. Prove that the circumcircle of $\tri... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 852f6968-92a8-564d-b585-3bccd1add7fc | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For positive real numbers $a,b,c$ with $a+b+c=1$, prove that
\[
\frac{a^4}{\sqrt{b^4+c^4}}+\frac{b^4}{\sqrt{c^4+a^4}}+\frac{c^4}{\sqrt{a^4+b^4}}\ge\frac{1}{3\sqrt2}\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 92491edd-13b9-5373-a2a4-d694f2381605 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n$ be a positive integer and let its positive divisors be listed in increasing order as $d_1=1<d_2<\dots<d_k=n$. Find all $n$ such that
1) $n = d_1^2+d_2^2+d_3^2+d_4^2+d_5^2+d_6^2$,
2) $n = d_1^2+d_2^2+d_3^2\,. $ | {
"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 | 87343b25-ab92-5351-9c83-d78a104afda6 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For positive real numbers $a,b,c$ with $abc=1$, prove the two inequalities
\[
(a+b)(b+c)(c+a)+7\ge 5(a+b+c)
\]
and
\[
a^{3}+b^{3}+c^{3}+15\ge 6\Bigl(\frac1a+\frac1b+\frac1c\Bigr).
\] | {
"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 | a56b3607-eab7-564b-8cd6-78caf9a21861 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Show that for every positive integer $n$, the set $\{1,2,\dots,2n\}$ can be partitioned into $n$ unordered pairs $\{a_i,b_i\}$ such that $a_i+b_i$ is a prime number for each $i$. | {
"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 | 3914eb10-adc5-5af6-b3fd-a13179e531b0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions \(f:\mathbb{R}\to\mathbb{R}\) such that for all real numbers \(x,y\),
\[
f(x)+f\bigl(x+f(y)\bigr)=y+f\bigl(f(x)+f\bigl(f(y)\bigr)\bigr).
\] | {
"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 | e0551722-520c-5ce1-acd1-4fd65411435b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $O$ be the circumcenter of $\triangle ABC$ with $AB<AC$. Let $D$ be a point on the internal angle bisector of $\angle BAC$. Let $E$ be the point on $BC$ such that $OE\parallel AD$ and $DE\perp BC$. Let $K$ be the point on the extension of $EB$ such that $EK=EA$. The circumcircle of $\triangle ADK$ meets $BC$ again ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 6c260fea-0a2a-5b6b-995a-d9d6e271a630 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $k$ be an integer with $k>1$. Prove that the Diophantine equation
\[
(x+1)(x+2)\cdots(x+k)=y^{2}
\]
has only finitely many integer solutions $(x,y)$. | {
"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 | 666b1e3e-7dab-568b-a735-0cde59b79282 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $T_n$ be the tournament on vertices $v_1,\dots,v_n$ in which the directed edge $v_i\to v_j$ is present exactly when $i<j$ (i.e., all edges point from the smaller index to the larger index). A directed graph is called **unipathic** if for every ordered pair of vertices $(x,y)$ there is at most one directed $x\to y$ ... | {
"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 | 3841ca09-8814-52b6-adc7-772b5c4f62d2 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Call a positive integer $n$ a “Downhill Number’’ if its base‑10 representation $\overline{a_k a_{k-1}\dots a_0}$ satisfies $a_k\ge a_{k-1}\ge\dots\ge a_0$. Call a polynomial $P(x)\in\mathbb R[x]$ a “Downhill Integer‑Value Polynomial’’ (DI‑VP) if $P(n)\in\mathbb Z$ for every Downhill Number $n$. Determine whether every ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 01382128-5c14-5991-8e9d-811a442f391f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with $AB<AC$, and let $\Gamma$ be its circumcircle. Let $D$, $E$, and $F$ be the points where the incircle of $ABC$ touches $BC$, $CA$, and $AB$, respectively. Let $R$ be the point on $EF$ such that $DR\perp EF$ (i.e., $DR$ is the altitude from $D$ in $\triangle DEF$). Let $S$ be the intersectio... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 9f13d2b1-53d7-5763-9b27-13299a3a1edb | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$,
\[
f\bigl(y\,f(x+1)+x^{2}\bigr)=x\,f(x+y)+f(y).
\] | {
"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 | be134977-61c0-5dcf-b5d8-13d99d97400c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(A\) be a set of positive integers containing at least two elements. Suppose that for any two distinct elements \(a,b\in A\) with \(a>b\) the integer
\[
\frac{[a,b]}{a-b}
\]
also belongs to \(A\), where \([a,b]\) denotes the least common multiple of \(a\) and \(b\). Prove that \(A\) contains exactly two elements. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | bb282d80-781a-577c-9999-3c890923de70 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c$ be the side lengths of a triangle $ABC$ with opposite angles $A,B,C$. Prove that
\[
4\cos A\cos B\cos C+\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\le 2\,.
\] | {
"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 | dfa37161-db6e-5824-b05e-cc45903cfcb1 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine all functions $f:\mathbb N\to\mathbb N$ that are eventually surjective (i.e., there exists $N\in\mathbb N$ such that for every $n\ge N$ there is $m\in\mathbb N$ with $f(m)=n$) and satisfy
\[
f(f(a)) + b \mid a^2 + f(a)f(b)
\]
for all $a,b\in\mathbb 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 | 781477f0-d682-5c60-a542-eb7637397f03 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Circles $\omega_1$ and $\omega_2$ have centers $O_1$ and $O_2$, respectively, and intersect at two points $X$ and $Y$. $AB$ is a common external tangent of the two circles, with $A\in\omega_1$ and $B\in\omega_2$. The tangents to $\omega_1$ and $\omega_2$ at $X$ meet the line $O_1O_2$ at $K$ and $L$, respectively. Le... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 62a47b6e-4ef2-5cc1-8e1a-3d504f8bfa6e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given a triangle $ABC$ with circumcircle $(O)$ and incenter $I$. Let $l_1$ and $l_2$ be two distinct lines through $I$. Let $A_1$ and $A_2$ be the second intersection points of $l_1$ and $l_2$, respectively, with the circumcircle of $\triangle BIC$. Define $B_1,B_2$ analogously as the second intersections of $l_1,l_2$ ... | {
"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 | c351845e-3fed-51c4-8fda-7c3b6907b99c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\triangle ABC$ be a triangle. Let $P$ and $Q$ be isogonal conjugates with respect to $\triangle ABC$, and let $R$ and $S$ be another isogonal conjugate pair with respect to $\triangle ABC$. Let $\mathcal C_{1}$ be the unique circumconic of $\triangle ABC$ that passes through $P$ and $R$, and let $X$ be the second ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | dd1966e0-3e9a-5ec0-a6b4-a5d4229aa94c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n$ be a positive integer. Consider the $n$‑dimensional hypercube $Q_n$, whose vertices are all binary strings of length $n$, with two vertices adjacent (neighbors) when they differ in exactly one coordinate. Initially $k$ vertices are colored black. In each step any uncolored vertex that has at least two black nei... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 8dc9ea47-c932-5350-8ef2-e69d7d034ee7 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | The circle inscribed in triangle $ABC$ touches the sides $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Let $X$, $Y$, $Z$ be the midpoints of $EF$, $FD$, $DE$, respectively. Prove that the incenter $I$ of $ABC$, the circumcenter $O$ of $ABC$, and the circumcenter of triangle $XYZ$ are collinear. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 50e71375-27bb-50ca-af2d-2dbe9cfeffe0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(a,b,c\in\mathbb{R}\). Prove that
\[
a^{4}+b^{4}+c^{4}+2\bigl(a^{3}b+b^{3}c+c^{3}a\bigr)+11\bigl(a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2}\bigr)\ge 6\bigl(ab^{3}+bc^{3}+ca^{3}\bigr)+8abc\,(a+b+c),
\]
and determine all equality cases. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 63383f3d-dd62-5ba7-a28d-c97e1a360af0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $O$ and $H$ be the circumcenter and orthocenter, respectively, of an acute triangle $ABC$. Choose points $D\in AB$ and $E\in AC$ such that $A,D,O,E$ are concyclic. Let $P$ be any point on the circumcircle of $ABC$. Through $P$ draw a line parallel to $OD$, meeting $AB$ at $X$, and a line parallel to $OE$, meeting $... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 643519b4-017c-59d4-8185-dee0e6c29631 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb R\to\mathbb R$ such that
\[
(1-f(x)f(y))\,f(x+y)=f(x)+f(y)\qquad\forall\,x,y\in\mathbb R.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | b6503e53-db26-52dc-bfc6-f4025cf7c3aa | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\ge 0$ be real numbers such that $a^{2}+b^{2}+c^{2}=3$. Prove
\[
2\bigl(a^{2}b+b^{2}c+c^{2}a\bigr)\le abc\,(a+b+c-1)+4.
\] | {
"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 | 5add4e38-0d10-567f-b850-903a17f973a8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let Ω and ω be two circles with ω inside Ω. Let points A, B, C, A₁, A₂, B₁, B₂, C₁, C₂ lie on Ω such that the segments AA₁, AA₂, BB₁, BB₂, CC₁, CC₂ are tangent to ω. Define
\(Z = A_1A_2 \cap B_1B_2,\qquad X = B_1B_2 \cap C_1C_2,\qquad Y = C_1C_2 \cap A_1A_2.\)
Prove that the lines \(AX,\; BY,\; CZ\) 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 | 6639ca4f-4de5-5529-bb1b-b8ec2d404093 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,\;y,\;z>0$ satisfy $xy+yz+zx=1$. Prove the following inequalities:
1. \((x^{2}y+y^{2}z+z^{2}x)(x+y+z)\ge 1\);
2. \((z^{2}y+zx^{2}+xy^{2})(x^{2}y+y^{2}z+z^{2}x)\ge \dfrac13\);
3. \(xy(x^{2}+y)+yz(y^{2}+z)+zx(z^{2}+x)\ge \dfrac13(x+y+z)+\dfrac13\);
4. \(\displaystyle\frac{4}{3}\Big(\sum_{\text{cyc}}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 | 8d5383c4-7671-599d-bacd-8e34e4093e3c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$,
$$
f\bigl(f(x)-y^{2}\bigr)=f(x)^{2}-2\,f(x)\,y^{2}+f\bigl(f(y)\bigr)\,.
$$ | {
"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 | 5438f5f7-0f82-5471-8814-cfb895458eac | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For positive real numbers $a,b,c$, prove that
\[
\sum_{\text{cyc}}\frac{a^{2}+b^{2}}{c\,(a+b)}\;\ge\;\Bigl(\sum_{\text{cyc}}a^{2}\Bigr)\sqrt{\frac{3}{abc\,(a+b+c)}}\,,
\]
where $\displaystyle\sum_{\text{cyc}}$ denotes the cyclic sum over $(a,b,c)$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | e3ac6701-9440-50f8-adaf-f82949b2e60b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let triangle $ABC$ have circumradius $R=1$. Denote by $r$ the inradius of $ABC$ and by $p$ the inradius of the orthic triangle $A'B'C'$ of $ABC$. Prove that
\[p\le 1-\frac13\,(1+r)^2.\] | {
"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 | 43aad0b4-92ca-554c-b912-d2dac0c8b830 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $k$ be a positive integer and let $r_1,r_2,\dots,r_k$ be positive real numbers such that for every integer $n\ge 2$ the sum
\[P(n)=\sum_{j=1}^{k} r_j^{1/n}\]
is a rational number. Prove that $r_1=r_2=\dots =r_k=1\,. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | e9b258ee-3ec3-5869-9fd2-948c511aefe6 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In a cyclic quadrilateral $ABCD$, let $E$ be the intersection of the extensions of $AB$ and $DC$ (i.e. $E=AB\cap DC$) and let $F$ be the intersection of the diagonals $AC$ and $BD$ (i.e. $F=AC\cap BD$). Let $P$ be a point on the ray $EF$ (the ray starting at $E$ and passing through $F$) such that $\angle BPE=\angle CPE... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 6e67fde9-2922-54a3-87a5-ca0403e9266b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle with circumcircle $\omega$ and let $\ell$ be a line that does not intersect $\omega$. Let $O$ be the centre of $\omega$ and let $P$ be the foot of the perpendicular from $O$ to $\ell$ (so $OP\perp\ell$). The extensions of the sides $BC$, $CA$, $AB$ meet $\ell$ at points $X$, $Y$, $Z$, respective... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 5a1fc34c-2aa5-5b11-ae40-cf33730056cc | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $H$ be the orthocenter of $\triangle ABC$ with $AB\neq AC$, and let $M$ be the midpoint of $BC$. Let $\ell$ be the line through $H$ perpendicular to $MH$, and denote by $E$ and $D$ the intersections of $\ell$ with $AB$ and $AC$, respectively. Let the circumcircles $(ABC)$ and $(AED)$ intersect again at $F\neq A$. 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 | 83ac8824-aaa2-54e3-9e77-6883a35ac971 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that for positive real numbers $a,b,c$,
\[
\Bigl(\frac1a+\frac1b+\frac1c\Bigr)\Bigl(\frac1{1+a}+\frac1{1+b}+\frac1{1+c}\Bigr)\ge\frac9{1+abc}\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 2de74190-54f0-5242-9b14-d1862b21744c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find the following functions:
1. All continuous functions \(f:(0,+\infty)\to(0,+\infty)\) satisfying
\[
f(x)\,f(y)=f(xy)+f\!\left(\frac{x}{y}\right)\qquad\forall\,x,y>0.
\]
2. All functions \(f:\mathbb{R}\to\mathbb{R}\) such that
\[
f(x+y)=f(x)+f(y)\qquad\forall\,x,y\in\mathbb{R},
\]
and
\[
f\!\left(\frac1x\ri... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 25c91b87-f92b-5231-9cd1-eda0ba821c82 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let A be the set of all integer sequences (a₁,a₂,a₃,…) with aᵢ∈ℤ. Let φ: A→ℤ be a function satisfying
(i) φ(s + t) = φ(s) + φ(t) for all s,t∈A, where (s + t)ᵢ = sᵢ + tᵢ (term‑wise addition);
(ii) for each n∈ℕ, φ(eₙ) = 0, where eₙ is the sequence that has a 1 in the n‑th position and 0 elsewhere.
(a) 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 | 1814fa16-47f6-5653-9d91-b19e4f8d490a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a quadrilateral with diagonals intersecting at $I=AC\cap BD$. Let $I_{1},I_{2},I_{3},I_{4}$ be the incenters of triangles $IAB$, $IBC$, $ICD$, and $IDA$, respectively. Prove that the quadrilateral $I_{1}I_{2}I_{3}I_{4}$ is cyclic if and only if $ABCD$ is tangential (i.e., $ABCD$ has an inscribed circle). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 75e15950-b909-50da-9ccc-cc00bba3d47a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine all functions \(f:\mathbb{R}^{+}\to\mathbb{R}^{+}\) such that
\[
f\bigl(x+f(x+y)\bigr)=f(2x)+y\qquad\text{for all }x,y\in\mathbb{R}^{+}.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 71632b7f-1f59-5828-8e7e-c61960e9bb43 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Given $n$ points in the plane with $n\ge4$, no three collinear, let $A(n)$ denote the number of parallelograms whose vertices are among these points and whose area equals 1. Prove that for all $n\ge4$
\[
A(n)\le\frac{n^{2}-3n}{4}\,.
\] | {
"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 | 6b38757f-3f0e-581a-b132-d19e10f404fc | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all positive integers \(x,\;y,\;m,\;n\) such that \(|x-y|=1\) and \(x^{m}-y^{n}=1\). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 6a407f58-3ea4-5f73-8162-9d5ad7dcb8d3 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all integers $a,b,c$ such that
\[
a^5+b^{1890}=c^{19}.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | fc4bea3c-7c6f-52a3-a1af-dc98bb9caf81 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be a triangle inscribed in its circumcircle $\Gamma$ with centre $O$. Let $I$ be the incenter of $ABC$, and let $D$, $E$, $F$ be the points where the incircle touches $BC$, $CA$, and $AB$, respectively. Let $S$ be the foot of the perpendicular from $D$ to the line $EF$. Prove that the line $SI$ passes through... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 7374338c-00f7-5b12-b762-2fcfffb883a9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Arnaldo and Bernaldo play the following game. They alternately say natural numbers, and the player who says 0 wins. After the first move, if the previous number is $n$, write its binary expansion
\[
n=\sum_{m\in O_n}2^m,
\]
where $O_n$ is the set of exponents that appear. The next player may say any $m\in O_n$.
Defin... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | b2d24a27-5c5d-55a5-a9c7-0cb08633dca9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\mathbb N$ denote the set of positive integers. Find all functions $f:\mathbb N\to\mathbb N$ such that
\[
\frac{f(x)-f(y)+x+y}{\,x-y+1\,}
\]
is an integer for every $x,y\in\mathbb N$ with $x>y$. | {
"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 | ac267b77-16a4-58cb-b1b0-1a27d6bc42e6 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,y,z\ge0$ be real numbers such that $x+y+z=3$. Prove that
$$xyz\bigl(xy^2+yz^2+zx^2\bigr)\;+\;3\bigl(x^2y+y^2z+z^2x\bigr)\le12.$$ | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 11b3d579-0105-536b-abbc-71eeb48bdfdc | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | An integer $n\ge 1$ is called **conservative** if the smallest prime divisor of $(n!)^{\,n}+1$ does not exceed $n+2015$. Decide whether there are infinitely many conservative 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 | 08e81c2e-cedd-5647-8fa1-353f75d9d4c9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Consider the Diophantine equation
\(x^2 + y^2 + z^2 + t^2 - N\,x y z t - N = 0,\)
where \(N\) is a positive integer and \(x, y, z, t\) are required to be positive integers.
(a) Prove that there exist infinitely many positive integers \(N\) for which the equation has at least one positive integer solution \((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 | b8d7dd91-01d2-533c-b389-53a407f37001 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an acute‑angled triangle with orthocenter $H$ and circumcircle $k$. Let $\omega$ be the circle with diameter $AH$, and let $P\neq A$ be the second intersection of $\omega$ with $k$. Let $Q\neq P$ be the second intersection of line $BP$ with $\omega$, and let $R\neq P$ be the second intersection of line $CP... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 0bd582a9-7ce1-5f18-85a2-03d4245ca651 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find the largest real number $c$ such that for every positive integer $n$ and for all real numbers $x_1,x_2,\dots,x_n$,
\[
\sum_{i=1}^{n}\sum_{j=1}^{n}\bigl(n-|i-j|\bigr)x_i x_j \;\ge\; c\sum_{i=1}^{n}x_i^{2}.
\] | {
"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 | fca53fbe-f2dc-54cb-957e-564ce9770b18 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
\[
f(x+y)=f(x)+f(y)\qquad\text{for all }x,y\in\mathbb{R},
\]
and
\[
f(x)\,f\!\bigl(\tfrac1x\bigr)=1\qquad\text{for all }x\neq0.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | bf471278-558a-5572-8115-2699af28b849 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine all functions $f:[0,\infty)\to\mathbb{R}$ such that
1. $\displaystyle\lim_{x\to+\infty}f(x)=0$,
2. $f(x)+3f(x^{2})=5\,f(x^{3}+1)$ for all $x\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 | fc4cba8c-94bf-5bd3-8404-e4cf5455c540 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $P$ be a point and let $D,E,F$ be the cevian triangle of $P$ with respect to $\triangle ABC$ (i.e. $D\in BC$, $E\in CA$, $F\in AB$ and $AD,BE,CF$ concur at $P$). Let $(DEF)$ be the circumcircle of $\triangle DEF$. Let $X,Y,Z$ be the second intersections of $(DEF)$ with $BC,CA,AB$ respectively (so $X\neq D$, $Y\neq ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | e513c94c-aa27-58e2-9053-4a3985371ab6 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $m,n$ and $a_1,a_2,\dots,a_m$ be arbitrary positive integers. Two players, Ali and Mohammad, play the following game. At each step Ali chooses numbers $b_1,b_2,\dots,b_m\in\mathbb N$. Mohammad then chooses a positive integer $s$ and produces a new sequence $\{c_i\}_{i=1}^m$ defined by
\[
c_i = a_i + b_{i+s},
\qqu... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 98b1fac8-39db-5c21-ba4d-b659e3b6f7c8 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that for positive real numbers $a,b,c$,
\[
\frac{8a^{2}+2ab}{\bigl(b+\sqrt{6ac}+3c\bigr)^{2}}
+\frac{2b^{2}+3bc}{\bigl(3c+\sqrt{2ab}+2a\bigr)^{2}}
+\frac{18c^{2}+6ac}{\bigl(2a+\sqrt{3bc}+b\bigr)^{2}}
\ge 1.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 7a81ff07-894a-597c-9238-d5e7c4d07ccd | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Prove that for any triangle with angles $A$, $B$, $C$ (so $A,B,C>0$ and $A+B+C=\pi$), the following inequality holds:
\[
\frac{\tan\frac{A}{2}}{\tan\frac{B}{2}}+\frac{\tan\frac{B}{2}}{\tan\frac{C}{2}}+\frac{\tan\frac{C}{2}}{\tan\frac{A}{2}}
\;\ge\;
\frac{\sin\frac{A}{2}}{\sin\frac{B}{2}}+\frac{\sin\frac{B}{2}}{\sin\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 | e70571eb-9cda-5664-bc01-885b40c29f8c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Fifty teams take part in a round‑robin tournament that lasts 50 days. On each day a non‑empty subset of the teams (at least two teams) shows up, and any two teams that are present on the same day play each other that day (i.e., the set of teams present on a day forms a complete graph). Moreover, every unordered pair 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 | 01148b2c-6ad6-53bd-9cb8-ff57109ee9cc | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x,y,z\ge0$ be real numbers satisfying $x+y+z=3$. Prove that
\[
\bigl(x^{3}+y^{3}+z^{3}\bigr)\bigl(x^{3}y^{3}+y^{3}z^{3}+z^{3}x^{3}\bigr)\le 36\,(xy+yz+zx)\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 438beb75-b022-5ed0-9975-1d0c77ba2247 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $(I)$ be the incircle of triangle $A_1A_2A_3$. For each $i\in\{1,2,3\}$, let $B_i$ be the point of tangency of the unique circle that passes through the other two vertices $A_j$ and $A_k$ (where $\{i,j,k\}=\{1,2,3\}$) and is tangent to $(I)$. Prove that the three lines $A_1B_1$, $A_2B_2$, $A_3B_3$ 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 | 382f79bc-fab4-5a3b-864a-f3a714411bd0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $BE$ and $CF$ be the altitudes of triangle $ABC$, with orthocenter $H$ (so $E\in AC$, $F\in AB$). Let $X=EF\cap BC$ (the intersection of line $EF$ with line $BC$, possibly on the extension). Let $Y$ be the second intersection of the circumcircles of $\triangle XEB$ and $\triangle XFC$. Let $Z$ be the second interse... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | e25fa891-5f6d-5a50-a8ef-4a6f89ea143e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\alpha$ and $\beta$ be rational multiples of $\pi$ that lie in the interval $(0,\frac{\pi}{2})$. Determine whether the ratio $\displaystyle\frac{\cos\alpha}{\cos\beta}$ can be a rational number. (The restriction to $(0,\frac{\pi}{2})$ is intended to avoid the obvious examples where the ratio equals $1$, $\frac12$,... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 403dab1d-c800-58f9-83e5-d16b933ecee1 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $x, y, z>0$ satisfy $x+y+z=x^{3}+y^{3}+z^{3}$. Prove that
\[
\frac{xz^{2}}{(x^{2}+1)\,y^{2}}+\frac{yx^{2}}{(y^{2}+1)\,z^{2}}+\frac{zy^{2}}{(z^{2}+1)\,x^{2}}\ge \frac{x+y+z}{2}\, .
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ea322066-a892-55a2-8484-86b7b8a292bd | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABC$ be an obtuse triangle with $\angle A>90^\circ$. Let $O$ be its circumcircle and let $AK$ be a diameter of $O$ (so $K$ is the antipode of $A$). Let $D$ be a point on segment $AB$ such that $AD=AC$. Let $L=AK\cap CD$. Let the circle through $D,K,L$ intersect $O$ again at $P\neq K$. Given that $AK=2$ and $\angle... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 2334c6a1-ed87-5b84-bbf0-f98e99539319 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find the largest real constant $C$ such that for every choice of $2019$ distinct positive real numbers $a_1,a_2,\dots,a_{2019}$ the inequality
\[
\frac{a_1}{|a_2-a_3|}+\frac{a_2}{|a_3-a_4|}+\cdots+\frac{a_{2019}}{|a_1-a_2|}>C
\]
holds. (Indices are taken modulo $2019$, i.e. $a_{2020}=a_1$, $a_{2021}=a_2$.) | {
"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 | 46f2afd9-74f6-5b9c-8535-23fc96455772 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}^{*}\to\mathbb{R}^{*}$ (where $\mathbb{R}^{*}=\mathbb{R}\setminus\{0\}$) such that
\[
f(x^{2}+y)=f(x^{2})+\frac{f(xy)}{f(x)}
\]
for all non‑zero real numbers $x$ and $y$ satisfying $y\neq -x^{2}$ (so that $x^{2}+y\neq0$). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | b31b772f-2855-5c4d-b51a-193f5816951c | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In a scalene triangle $ABC$, let $I$ be its incenter, $N_a$ its Nagel point, and $J$ the symmedian point of the excentral triangle $I_aI_bI_c$. Let $D,E,F$ be the midpoints of $BC,CA,AB$, and let $X,Y,Z$ be the points where the incircle $(I)$ touches $BC,CA,AB$, respectively. From $X,Y,Z$ draw the perpendiculars to the... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | e18de50f-11e9-5228-9f0f-998d5f4c835d | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $R_1,R_2,\dots,R_m$ be a collection of subsets (routes) of a finite set $S$ (stops) such that
1. $|R_i|\ge 4$ for every $i$, and
2. $|R_i\cap R_j|=1$ for every distinct $i,j$.
Prove that there exists a partition $S=S_1\cup S_2$, $S_1\cap S_2=\varnothing$, (equivalently, a 2‑coloring of the stops) such 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 | dab7047f-8e44-5a1b-9023-0c3601abe848 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
\[
f\!\left(\frac{x\,f(y)}{2}\right)+f\!\left(\frac{y\,f(x)}{2}\right)=4xy
\qquad\text{for all real }x,y.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 3e64ea27-1822-5bc2-b095-728275179f74 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $p(x)\in\mathbb Z[x]$ be a polynomial with integer coefficients such that $p(n)>n$ for every positive integer $n$. Define a sequence by $x_1=1$ and $x_{k+1}=p(x_k)$ for $k\ge1$. Assume that for every positive integer $N$ there exists some term of the sequence that is divisible by $N$ (i.e. $\exists\,k\ge1$ with $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 | 5176c952-0f6a-596a-b8a2-b937bfc19a1f | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a cyclic quadrilateral. Points $P,Q,R,S$ lie on the sides $AB,BC,CD,DA$ respectively such that
- $AP=AS=CR=CQ$,
- $PQ\parallel RS$,
- $\displaystyle\angle APS+\angle CRQ=\angle BPQ+\angle DRS$.
Prove that
a) $\operatorname{area}(BPSD)=\operatorname{area}(BQRD)$;
b) $\displaystyle\frac{1}{9... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ae553dd7-e9ca-5a17-bdff-7cef315184c9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | All the squares of a rectangular board with \((n+1)\) columns and \((n-1)\) rows are painted using three colors. The coloring must satisfy the condition that for any two distinct columns and any two distinct rows, the four squares at their intersections are **not** all the same color. Determine the greatest 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 | 6341e850-f85c-58f7-be50-88c7c66adf53 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | For positive real numbers $a,b,c,d,e$ satisfying $abcde=1$, prove that
\[
\sum_{\text{cyc}}\frac{1}{\sqrt{a^{4}+2b^{4}+3c^{4}+4d^{4}+10}}\le\frac{\sqrt5}{2}\,,
\]
where the cyclic sum runs over the five ordered 5‑tuples $(a,b,c,d,e),\,(b,c,d,e,a),\,(c,d,e,a,b),\,(d,e,a,b,c),\,(e,a,b,c,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 | ad528a83-979f-51b5-af38-10d152149b3a | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Consider an infinite rectangular grid of cells (the integer lattice) where each cell contains a lamp that is initially turned off. An allowed operation consists of selecting a square subgrid of size \(3\times3\), \(4\times4\), or \(5\times5\) cells and toggling every lamp in that square (changing on to off and off to o... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 213cc84b-a7cf-500e-acb7-2a814eebce6e | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $\triangle ABC$ be an acute triangle. Denote by $D$, $E$, $F$ the feet of the altitudes from $A$, $B$, $C$ onto the sides $BC$, $CA$, $AB$, respectively. Let $P$ be one of the intersection points of line $EF$ with the circumcircle of $\triangle ABC$. Let $Q$ be the intersection of lines $BP$ and $DF$. Prove that $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 | 2184f06d-4612-5e10-a00d-9588995be2b9 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\in\mathbb{R}^+$ . Prove that
\[
\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^{2}\ge (a+b+c)\left(\frac1a+\frac1b+\frac1c\right)+\frac{(a-b)^{2}}{ab+bc+ca}\,.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | f61cb4ea-2546-5189-a040-3d984ffc1a78 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $m,n$ be positive integers and define
\[
S=\{(x,y)\in\mathbb Z^2\mid m\le x^2+y^2\le n\}.
\]
For every line $\ell$ in the plane that is either horizontal ($y=c$), vertical ($x=c$), or has slope $+1$ ($x-y=c$) or slope $-1$ ($x+y=c$), Alice tells Bob the number of points of $S$ lying on $\ell$. Prove that from ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 6fabe9fc-f2df-52f3-8d47-408046727915 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In a trapezoid \(ABCD\) with bases \(AD\parallel BC\), let the diagonals intersect at \(O\). Given that \(\angle AOB = 60^\circ\), prove (or disprove) that
\[
AD \le AB + CD.
\] | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 235b6a66-5857-5718-909b-bb3e9e4b1655 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c\ge 0$. Prove that
\[
\sqrt{a^{2}+bc}+\sqrt{b^{2}+ca}+\sqrt{c^{2}+ab}\le\frac32\,(a+b+c).
\]
Is it true that the stronger inequality
\[
\sqrt{a^{2}+bc}+\sqrt{b^{2}+ca}+\sqrt{c^{2}+ab}\le\sqrt2\,(a+b+c)
\]
holds for all non‑negative $a,b,c$? | {
"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 | e8f13dfb-e22d-5547-a3e5-ca4b293a230b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine all functions $f:\mathbb{R}\to\mathbb{R}^{+}$ such that for all real numbers $x,y$ the following conditions hold:
(i) $f(x^{2})=f(x)^{2}-2xf(x)$;
(ii) $f(-x)=f(x-1)$;
(iii) $1<x<y\implies f(x)<f(y)\,$ (i.e., $f$ is strictly increasing 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 | b9aea4cc-9884-59c7-b650-a471b171bb34 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $n>1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k)_{k\ge 1}$, defined by
\[
a_k=\Big\lfloor\frac{n^{k}}{k}\Big\rfloor,
\]
are odd. (Here $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$.) | {
"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 | 03cb7436-c3e8-51ce-9757-0cd3d507a2eb | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In triangle $ABC$ let $I$ be the incenter, $L$ be the midpoint of arc $BAC$ on the circumcircle, $\omega$ be the incircle, and $\gamma$ be the circle with diameter $\overline{AI}$. Let $\omega$ be tangent to $\overline{BC}$ at $D$, and let the ray $\overline{DI}$ intersect $\omega$ again at $X$ and intersect $\gamma$ ... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 30044e6c-3d42-5ea1-ba71-25a85db61e7b | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $a,b,c>0$. Prove that
\[
\frac{ab}{a^2+ab+b^2}+\frac{bc}{b^2+bc+c^2}+\frac{ca}{c^2+ca+a^2}
\;+\;
\frac{4(a^2+b^2+c^2)}{(a+b+c)^2}
\;\ge\;\frac73.
\]
(The sum $\sum$ is taken cyclically over $(a,b,c)$.) | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | fbf592bf-333e-52a8-aac3-291356b527a4 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find the smallest positive integer that is a multiple of 2016 and whose sum of its decimal digits equals 2016. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 6304b7c7-8f54-573b-a161-53fba389bf83 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Find all functions \(g:\mathbb N\to\mathbb N\) such that
\[
\bigl(g(m)+n\bigr)\bigl(g(n)+m\bigr)
\]
is a perfect square for every pair of natural numbers \(m,n\) (where \(\mathbb N=\{1,2,3,\dots\}\)). | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 48850f01-d00a-5d42-a30e-abbe2cad24f5 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | There is a new chess piece, the Prince, that moves one square either vertically or horizontally (no diagonal moves). Does there exist a closed path on an 8 × 8 chessboard that visits every square exactly once (starting and ending on the same square) and uses the same number of horizontal moves as vertical moves? | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | f66a6ab1-7511-5c6a-a5f5-1636233d66a0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine a planar figure that has the same perimeter as an ellipse with semi‑axes $a>b>0$, and whose area exceeds the area of that ellipse by exactly $(a-b)^2$. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 9420206f-eb83-56c7-8a47-e94e70c69796 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let $ABCD$ be a quadrilateral with $AB=CD$.
Let $P$ be the intersection of the internal bisectors of $\angle A$ and $\angle B$, and let $Q$ be the intersection of the internal bisectors of $\angle C$ and $\angle D$.
Let $E$ and $F$ be the midpoints of $BC$ and $DA$, respectively.
Let $M$ be the intersection of $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 | aa60ee18-154c-5a3d-b0fc-98366a2caec0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | If we are given the side lengths $a=BC$ and $b=AC$ of a triangle $ABC$ and the difference of its base angles $\beta-\gamma=\angle ABC-\angle ACB$, can the triangle $ABC$ be constructed with straightedge and compass? | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | ccfadad0-6a6a-5181-b17d-32b897e63826 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | In triangle $ABC$ with incenter $I$, let $M$ be the midpoint of $BC$. Let $D=IM\cap AB$ and $E=IM\cap AC$. Let $F$ be the second intersection of the circumcircles of $\triangle ADE$ and $\triangle ABC$ (i.e., the point other than $A$ common to both circles). Let $G\neq F$ be the second intersection of line $FM$ with 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 | bfd594c1-47a9-54eb-b345-f340afa20ec0 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Let \(P\) be a convex polyhedron all of whose faces are triangles. Prove that the edges of \(P\) can be colored black and white so that for any two vertices of \(P\) there exists a path consisting only of black edges joining them, and also a path consisting only of white edges joining them. | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | 040620e9-903c-558f-8fc6-117702956a25 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Determine all integer pairs \((x,y)\) such that \(x(x-2)\) is a perfect cube; equivalently solve the Diophantine equation
\[
x(x-2)=y^{3}\,,
\]
with \(x,y\in\mathbb Z\). | {
"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 | c54179b1-df86-5078-a442-4420251cd5d6 | [] | AoPS | Nemotron-Math-Proofs-v3 |
{
"name": "proof_simple_agent"
} | Consider a circle S and an external point A. Let B and C be the points of tangency from A to S, i.e. AB and AC are tangent to S. Let D and E be points on S such that A, D, E are collinear in this order. Let F = BC ∩ AD and let M be the midpoint of BC. Suppose BF > FC. Prove that
(a) BM² = DM·ME,
(b) ∠DME = 2·∠DBE... | {
"input": [
{
"role": "user",
"content": "Your task is to solve a given problem. The problem may ask you to prove a statement, or ask for an answer. If finding an answer is required, you should come up with the answer, and your final solution should also be a rigorous proof of that answer being valid... | cc-by-4.0 | dfe17ff2-9b0a-50f0-9a6c-379a7069a08c | [] | AoPS | Nemotron-Math-Proofs-v3 |
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