Found problems: 85335
2014 Contests, 1
Determine all triples $(a,b,c)$, where $a, b$, and $c$ are positive integers that satisfy
$a \le b \le c$ and $abc = 2(a + b + c)$.
2014 Saudi Arabia BMO TST, 3
Let $ABCD$ be a parallelogram. A line $\ell$ intersects lines $AB,~ BC,~ CD, ~DA$ at four different points $E,~ F,~ G,~ H,$ respectively. The circumcircles of triangles $AEF$ and $AGH$ intersect again at $P$. The circumcircles of triangles $CEF$ and $CGH$ intersect again at $Q$. Prove that the line $P Q$ bisects the diagonal $BD$.
PEN A Problems, 68
Suppose that $S=\{a_{1}, \cdots, a_{r}\}$ is a set of positive integers, and let $S_{k}$ denote the set of subsets of $S$ with $k$ elements. Show that \[\text{lcm}(a_{1}, \cdots, a_{r})=\prod_{i=1}^{r}\prod_{s\in S_{i}}\gcd(s)^{\left((-1)^{i}\right)}.\]
1998 Mexico National Olympiad, 1
A number is called lucky if computing the sum of the squares of its digits and repeating this operation sufficiently many times leads to number $1$. For example, $1900$ is lucky, as $1900 \to 82 \to 68 \to 100 \to 1$. Find infinitely many pairs of consecutive numbers each of which is lucky.
2014 NIMO Summer Contest, 7
Evaluate \[ \frac{1}{729} \sum_{a=1}^{9} \sum_{b=1}^9 \sum_{c=1}^9 \left( abc+ab+bc+ca+a+b+c \right). \][i]Proposed by Evan Chen[/i]
2016 NIMO Problems, 7
Let $A$ and $B$ be points with $AB=12$. A point $P$ in the plane of $A$ and $B$ is $\textit{special}$ if there exist points $X, Y$ such that
[list]
[*]$P$ lies on segment $XY$,
[*]$PX : PY = 4 : 7$, and
[*]the circumcircles of $AXY$ and $BXY$ are both tangent to line $AB$.
[/list]
A point $P$ that is not special is called $\textit{boring}$.
Compute the smallest integer $n$ such that any two boring points have distance less than $\sqrt{n/10}$ from each other.
[i]Proposed by Michael Ren[/i]
2020 Hong Kong TST, 3
Two circles $\Gamma$ and $\Omega$ intersect at two distinct points $A$ and $B$. Let $P$ be a point on $\Gamma$. The tangent at $P$ to $\Gamma$ meets $\Omega$ at the points $C$ and $D$, where $D$ lies between $P$ and $C$, and $ABCD$ is a convex quadrilateral. The lines $CA$ and $CB$ meet $\Gamma$ again at $E$ and $F$ respectively. The lines $DA$ and $DB$ meet $\Gamma$ again at $S$ and $T$ respectively. Suppose the points $P,E,S,F,B,T,A$ lie on $\Gamma$ in this order. Prove that $PC,ET,SF$ are parallel.
2019 Romania EGMO TST, P2
Determine the digits $0\leqslant c\leqslant 9$ such that for any positive integer $k{}$ there exists a positive integer $n$ such that the last $k{}$ digits of $n^9$ are equal to $c{}.$
2021 Albanians Cup in Mathematics, 6
Let $p$ be an odd prime, and put $N=\frac{1}{4} (p^3 -p) -1.$ The numbers $1,2, \dots, N$ are painted arbitrarily in two colors, red and blue. For any positive integer $n \leqslant N,$ denote $r(n)$ the fraction of integers $\{ 1,2, \dots, n \}$ that are red.
Prove that there exists a positive integer $a \in \{ 1,2, \dots, p-1\}$ such that $r(n) \neq a/p$ for all $n = 1,2, \dots , N.$
[I]Netherlands[/i]
1992 India National Olympiad, 6
Let $f(x)$ be a polynomial in $x$ with integer coefficients and suppose that for five distinct integers $a_1, \ldots, a_5$ one has $f(a_1) = f(a_2) = \ldots = f(a_5) = 2$. Show that there does not exist an integer $b$ such that $f(b) = 9$.
2015 Online Math Open Problems, 28
Find the number of ordered pairs $(P(x),Q(x))$ of polynomials with integer coefficients such that
\[
P(x)^2+Q(x)^2=\left(x^{4096}-1\right)^2.
\]
[i]Proposed by Michael Kural[/i]
2017 Romania National Olympiad, 4
Let $a, b, c, d \in [0, 1]$. Prove that
$$\frac{a}{1 + b}+\frac{b}{1 + c}+\frac{c}{1 + d}+\frac{d}{1 + a}+ abcd \le 3.$$
2002 India National Olympiad, 6
The numbers $1, 2, 3$, $\ldots$, $n^2$ are arranged in an $n\times n$ array, so that the numbers in each row increase from left to right, and the numbers in each column increase from top to bottom. Let $a_{ij}$ be the number in position $i, j$. Let $b_j$ be the number of possible values for $a_{jj}$. Show that \[ b_1 + b_2 + \cdots + b_n = \frac{ n(n^2-3n+5) }{3} . \]
2005 Iran Team Selection Test, 2
Suppose there are $n$ distinct points on plane. There is circle with radius $r$ and center $O$ on the plane. At least one of the points are in the circle. We do the following instructions. At each step we move $O$ to the baricenter of the point in the circle. Prove that location of $O$ is constant after some steps.
2019 Vietnam National Olympiad, Day 2
Let $ABC$ be an acute, nonisosceles triangle with inscribe in a circle $(O)$ and has orthocenter $H$. Denote $M,N,P$ as the midpoints of sides $BC,CA,AB$ and $D,E,F$ as the feet of the altitudes from vertices $A,B,C$ of triangle $ABC$. Let $K$ as the reflection of $H$ through $BC$. Two lines $DE,MP$ meet at $X$; two lines $DF,MN$ meet at $Y$.
a) The line $XY$ cut the minor arc $BC$ of $(O)$ at $Z$. Prove that $K,Z,E,F$ are concyclic.
b) Two lines $KE,KF$ cuts $(O)$ second time at $S,T$. Prove that $BS,CT,XY$ are concurrent.
2016 Switzerland Team Selection Test, Problem 4
Find all integers $n \geq 1$ such that for all $x_1,...,x_n \in \mathbb{R}$ the following inequality is satisfied
$$\left(\frac{x_1^n+...+x_n^n}{n}-x_1....x_n\right)\left(x_1+...+x_n\right) \geq 0$$
2004 Germany Team Selection Test, 2
Let $n \geq 5$ be a given integer. Determine the greatest integer $k$ for which there exists a polygon with $n$ vertices (convex or not, with non-selfintersecting boundary) having $k$ internal right angles.
[i]Proposed by Juozas Juvencijus Macys, Lithuania[/i]
1983 Swedish Mathematical Competition, 4
$C$, $C'$ are concentric circles with radii $R$, $R'$. A rectangle has two adjacent vertices on $C$ and the other two vertices on $C'$. Find its sides if its area is as large as possible.
2014 Flanders Math Olympiad, 3
Let $PQRS$ be a quadrilateral with $| P Q | = | QR | = | RS |$, $\angle Q= 110^o$ and $\angle R = 130^o$ . Determine $\angle P$ and $\angle S$ .
2009 Postal Coaching, 2
Determine, with proof, all the integer solutions of the equation $x^3 + 2y^3 + 4z^3 - 6xyz = 1$.
2012 Czech-Polish-Slovak Match, 2
Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying
\[f(x+f(y))-f(x)=(x+f(y))^4-x^4\]
for all $x,y \in \mathbb{R}$.
2012 Kosovo National Mathematical Olympiad, 4
The right triangle $ABC$ with a right angle at $C$. From all the rectangles $CA_1MB_1$, where $A_1\in BC, M\in AB$ and $B_1\in AC$ which one has the biggest area?
2022 CMIMC, 4
Let $\triangle ABC$ be equilateral with integer side length. Point $X$ lies on $\overline{BC}$ strictly between $B$ and $C$ such that $BX<CX$. Let $C'$ denote the reflection of $C$ over the midpoint of $\overline{AX}$. If $BC'=30$, find the sum of all possible side lengths of $\triangle ABC$.
[i]Proposed by Connor Gordon[/i]
2018 IOM, 6
The incircle of a triangle $ABC$ touches the sides $BC$ and $AC$ at points $D$ and $E$, respectively. Suppose $P$ is the point on the shorter arc $DE$ of the incircle such that $\angle APE = \angle DPB$. The segments $AP$ and $BP$ meet the segment $DE$ at points $K$ and $L$, respectively. Prove that $2KL = DE$.
[i]Dušan Djukić[/i]
2011 Brazil National Olympiad, 2
33 friends are collecting stickers for a 2011-sticker album. A distribution of stickers among the 33 friends is incomplete when there is a sticker that no friend has. Determine the least $m$ with the following property: every distribution of stickers among the 33 friends such that, for any two friends, there are at least $m$ stickers both don't have, is incomplete.