Found problems: 85335
2020 Taiwan TST Round 2, 1
Find all triples $(a, b, c)$ of positive integers such that $a^3 + b^3 + c^3 = (abc)^2$.
2005 Moldova Team Selection Test, 2
Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$.
[i]Proposed by Hojoo Lee, Korea[/i]
2023 IFYM, Sozopol, 2
Find all functions $f: \mathbb{Z} \to \mathbb{Z}$ such that
\[
f(x) + f(y - 1) + f(f(y - f(x))) = 1
\]
for all integers $x$ and $y$.
1956 AMC 12/AHSME, 1
The value of $ x \plus{} x(x^x)$ when $ x \equal{} 2$ is:
$ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 16 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 36 \qquad\textbf{(E)}\ 64$
2014 Saint Petersburg Mathematical Olympiad, 3
$D$ is inner point of triangle $ABC$. $E$ is on $BD$ and $CE=BD$. $\angle ABD=\angle ECD=10,\angle BAD=40,\angle CED=60$ Prove, that $AB>AC$
2017 Brazil Team Selection Test, 3
Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$.
[i]Proposed by Evan Chen, Taiwan[/i]
2000 Slovenia National Olympiad, Problem 3
A point $D$ is taken inside an isosceles triangle $ABC$ with base $AB$ and $\angle C=80^\circ$ such that $\angle DAB=10^\circ$ and $\angle DBA=20^\circ$. Compute $\angle ACD$.
2007 Czech-Polish-Slovak Match, 2
The Fibonacci sequence is defined by $a_1=a_2=1$ and $a_{k+2}=a_{k+1}+a_k$ for $k\in\mathbb N.$ Prove that for any natural number $m,$ there exists an index $k$ such that $a_k^4-a_k-2$ is divisible by $m.$
2023 Argentina National Olympiad, 1
Let $n$ be a positive with $n\geq 3$. Consider a board of $n \times n$ boxes. In each step taken the colors of the $5$ boxes that make up the figure bellow change color (black boxes change to white and white boxes change to black)
The figure can be rotated $90°, 180°$ or $270°$.
Firstly, all the boxes are white.Determine for what values of $n$ it can be achieved, through a series of steps, that all the squares on the board are black.
2017 CMIMC Number Theory, 9
Find the smallest prime $p$ for which there exist positive integers $a,b$ such that
\[
a^{2} + p^{3} = b^{4}.
\]
2013 ISI Entrance Examination, 4
In a badminton tournament, each of $n$ players play all the other $n-1$ players. Each game results in either a win, or a loss. The players then write down the names of those whom they defeated, and also of those who they defeated. For example, if $A$ beats $B$ and $B$ beats $C,$ then $A$ writes the names of both $B$ and $C$. Show that there will be one person, who has written down the names of all the other $n-1$ players.
[hide="Clarification"]
Consider a game between $A,B,C,D,E,F,G$ where $A$ defeats $B$ and $C$ and $B$ defeats $E,F$, $C$ defeats $E.$ Then $A$'s list will have $(B,C,E,F)$, and will not include $G.$
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1989 IMO Shortlist, 31
Let $ a_1 \geq a_2 \geq a_3 \in \mathbb{Z}^\plus{}$ be given and let N$ (a_1, a_2, a_3)$ be the number of solutions $ (x_1, x_2, x_3)$ of the equation
\[ \sum^3_{k\equal{}1} \frac{a_k}{x_k} \equal{} 1.\]
where $ x_1, x_2,$ and $ x_3$ are positive integers. Prove that \[ N(a_1, a_2, a_3) \leq 6 a_1 a_2 (3 \plus{} ln(2 a_1)).\]
2007 Estonia National Olympiad, 2
A 3-dimensional chess board consists of $ 4 \times 4 \times 4$ unit cubes. A rook can step from any unit cube K to any other unit cube that has a common face with K. A bishop can step from any unit cube K to any other unit cube that has a common edge with K, but does not have a common face. One move of both a rook and a bishop consists of an arbitrary positive number of consecutive steps in the same direction. Find the average number of possible moves for either piece, where the average is taken over all possible starting cubes K.
2018 India IMO Training Camp, 1
Let $ABC$ be a triangle and $AD,BE,CF$ be cevians concurrent at a point $P$. Suppose each of the quadrilaterals $PDCE,PEAF$ and $PFBD$ has both circumcircle and incircle. Prove that $ABC$ is equilateral and $P$ coincides with the center of the triangle.
2021 Canadian Mathematical Olympiad Qualification, 6
Show that $(w, x, y, z)=(0,0,0,0)$ is the only integer solution to the equation
$$w^{2}+11 x^{2}-8 y^{2}-12 y z-10 z^{2}=0$$
2016 PUMaC Geometry A, 1
Let $\vartriangle ABC$ be an equilateral triangle with side length $1$ and let $\Gamma$ the circle tangent to $AB$ and $AC$ at $B$ and $C$, respectively. Let $P$ be on side $AB$ and $Q$ be on side $AC$ so that $PQ // BC$, and the circle through $A, P$, and $Q$ is tangent to $\Gamma$ . If the area of $\vartriangle APQ$ can be written in the form $\frac{\sqrt{a}}{b}$ for positive integers $a$ and $b$, where $a$ is not divisible by the square of any prime, fi nd $a + b$.
III Soros Olympiad 1996 - 97 (Russia), 10.1
Find the smallest natural number $n$ for which the equality $\sin n^o= \sin (1997n)^o$ holds.
1981 Kurschak Competition, 2
Let $n > 2$ be an even number. The squares of an $n\times n$ chessboard are coloured with $\frac12 n^2$ colours in such a way that every colour is used for colouring exactly two of the squares. Prove that one can place $n$ rooks on squares of $n$ different colours such that no two of the rooks can take each other.
1973 Chisinau City MO, 70
The natural numbers $p, q$ satisfy the relation $p^p + q^q = p^q + q^p$. Prove that $p = q$.
1994 All-Russian Olympiad, 1
Let be given three quadratic polynomials:
$P_1(x) = x^2 + p_1x+q_1, P_2(x) = x^2+ p_2x+q_2, P_3(x) = x^2 + p_3x+q_3$.
Prove that the equation $|P_1(x)|+|P_2(x)| = |P_3(x)|$ has at most eight real roots.
2013 European Mathematical Cup, 2
Let $P$ be a point inside a triangle $ABC$. A line through $P$ parallel to $AB$ meets $BC$ and $CA$ at points $L$ and $F$, respectively. A line through $P$ parallel to $BC$ meets $CA$ and $BA$ at points $M$ and $D$ respectively, and a line through $P$ parallel to $CA$ meets $AB$ and $BC$ at points $N$ and $E$ respectively. Prove
\begin{align*}
[PDBL] \cdot [PECM] \cdot [PFAN]=8\cdot [PFM] \cdot [PEL] \cdot [PDN] \\ \end{align*}
[i]Proposed by Steve Dinh[/i]
2025 Polish MO Finals, 4
A positive integer $n\geq 2$ and a set $S$ consisting of $2n$ disting positive integers smaller or equal to $n^2$ are given. Prove that there exists a positive integer $r\in \{1, 2, ..., n\}$ that can be written in the form $r=a-b$, for $a, b\in \mathbb{S}$ in at least $3$ different ways.
2016 Fall CHMMC, 6
How many binary strings of length $10$ do not contain the substrings $101$ or $010$?
1964 AMC 12/AHSME, 37
Given two positive number $a$, $b$ such that $a<b$. Let A.M. be their arithmetic mean and let G.M. be their positive geometric mean. Then A.M. minus G.M. is always less than:
$\textbf{(A) }\dfrac{(b+a)^2}{ab}\qquad\textbf{(B) }\dfrac{(b+a)^2}{8b}\qquad\textbf{(C) }\dfrac{(b-a)^2}{ab}$
$\textbf{(D) }\dfrac{(b-a)^2}{8a}\qquad \textbf{(E) }\dfrac{(b-a)^2}{8b}$
2006 Miklós Schweitzer, 7
Suppose that the function $f: Z \to Z$ can be written in the form $f = g_1+...+g_k$ , where $g_1,. . . , g_k: Z \to R$ are real-valued periodic functions, with period $a_1,...,a_k$. Does it follow that f can be written in the form $f = h_1 +. . + h_k$ , where $h_1,. . . , h_k: Z \to Z$ are periodic functions with integer values, also with period $a_1,...,a_k$?