Found problems: 96
2017 Turkey EGMO TST, 3
For all positive real numbers $x,y,z$ satisfying the inequality $$\frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}\leq 3,$$ prove that
$$\frac{x^2}{y^3}+\frac{y^2}{z^3}+\frac{z^2}{x^3}\geq \frac{x}{y}+\frac{y}{z}+\frac{z}{x}.$$
2021 EGMO, 1
The number 2021 is fantabulous. For any positive integer $m$, if any element of the set $\{m, 2m+1, 3m\}$ is fantabulous, then all the elements are fantabulous. Does it follow that the number $2021^{2021}$ is fantabulous?
2017 Turkey EGMO TST, 1
Let $m,k,n$ be positive integers. Determine all triples $(m,k,n)$ satisfying the following equation:
$3^m5^k=n^3+125$
2015 EGMO, 3
Let $n, m$ be integers greater than $1$, and let $a_1, a_2, \dots, a_m$ be positive integers not greater than $n^m$. Prove that there exist positive integers $b_1, b_2, \dots, b_m$ not greater than $n$, such that \[ \gcd(a_1 + b_1, a_2 + b_2, \dots, a_m + b_m) < n, \] where $\gcd(x_1, x_2, \dots, x_m)$ denotes the greatest common divisor of $x_1, x_2, \dots, x_m$.
2017 EGMO, 3
There are $2017$ lines in the plane such that no three of them go through the same point. Turbo the snail sits on a point on exactly one of the lines and starts sliding along the lines in the following fashion: she moves on a given line until she reaches an intersection of two lines. At the intersection, she follows her journey on the other line turning left or right, alternating her choice at each intersection point she reaches. She can only change direction at an intersection point. Can there exist a line segment through which she passes in both directions during her journey?
2016 EGMO, 4
Two circles $\omega_1$ and $\omega_2$, of equal radius intersect at different points $X_1$ and $X_2$. Consider a circle $\omega$ externally tangent to $\omega_1$ at $T_1$ and internally tangent to $\omega_2$ at point $T_2$. Prove that lines $X_1T_1$ and $X_2T_2$ intersect at a point lying on $\omega$.
2012 EGMO, 2
Let $n$ be a positive integer. Find the greatest possible integer $m$, in terms of $n$, with the following property: a table with $m$ rows and $n$ columns can be filled with real numbers in such a manner that for any two different rows $\left[ {{a_1},{a_2},\ldots,{a_n}}\right]$ and $\left[ {{b_1},{b_2},\ldots,{b_n}} \right]$ the following holds: \[\max\left( {\left| {{a_1} - {b_1}} \right|,\left| {{a_2} - {b_2}} \right|,...,\left| {{a_n} - {b_n}} \right|} \right) = 1\]
[i]Poland (Tomasz Kobos)[/i]
2017 EGMO, 2
Find the smallest positive integer $k$ for which there exists a colouring of the positive integers $\mathbb{Z}_{>0}$ with $k$ colours and a function $f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$ with the following two properties:
$(i)$ For all positive integers $m,n$ of the same colour, $f(m+n)=f(m)+f(n).$
$(ii)$ There are positive integers $m,n$ such that $f(m+n)\ne f(m)+f(n).$
[i]In a colouring of $\mathbb{Z}_{>0}$ with $k$ colours, every integer is coloured in exactly one of the $k$ colours. In both $(i)$ and $(ii)$ the positive integers $m,n$ are not necessarily distinct.[/i]
2017 Turkey EGMO TST, 5
In a $12\times 12$ square table some stones are placed in the cells with at most one stone per cell. If the number of stones on each line, column, and diagonal is even, what is the maximum number of the stones?
[b]Note[/b]. Each diagonal is parallel to one of two main diagonals of the table and consists of $1,2\ldots,11$ or $12$ cells.
2014 EGMO, 3
We denote the number of positive divisors of a positive integer $m$ by $d(m)$ and the number of distinct prime divisors of $m$ by $\omega(m)$. Let $k$ be a positive integer. Prove that there exist infinitely many positive integers $n$ such that $\omega(n) = k$ and $d(n)$ does not divide $d(a^2+b^2)$ for any positive integers $a, b$ satisfying $a + b = n$.
2014 EGMO, 2
Let $D$ and $E$ be points in the interiors of sides $AB$ and $AC$, respectively, of a triangle $ABC$, such that $DB = BC = CE$. Let the lines $CD$ and $BE$ meet at $F$. Prove that the incentre $I$ of triangle $ABC$, the orthocentre $H$ of triangle $DEF$ and the midpoint $M$ of the arc $BAC$ of the circumcircle of triangle $ABC$ are collinear.
2017 EGMO, 3
Let $n\geq1$ be an integer and let $t_1<t_2<\dots<t_n$ be positive integers. In a group of $t_n+1$ people, some games of chess are played. Two people can play each other at most once. Prove that it is possible for the following two conditions to hold at the same time:
(i) The number of games played by each person is one of $t_1,t_2,\dots,t_n$.
(ii) For every $i$ with $1\leq i\leq n$, there is someone who has played exactly $t_i$ games of chess.
2015 EGMO, 6
Let $H$ be the orthocentre and $G$ be the centroid of acute-angled triangle $ABC$ with $AB\ne AC$. The line $AG$ intersects the circumcircle of $ABC$ at $A$ and $P$. Let $P'$ be the reflection of $P$ in the line $BC$. Prove that $\angle CAB = 60$ if and only if $HG = GP'$
2012 EGMO, 3
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f\left( {yf(x + y) + f(x)} \right) = 4x + 2yf(x + y)\] for all $x,y\in\mathbb{R}$.
[i]Netherlands (Birgit van Dalen)[/i]
2024 EGMO, 3
We call a positive integer $n{}$ [i]peculiar[/i] if, for any positive divisor $d{}$ of $n{}$ the integer $d(d + 1)$ divides $n(n + 1).$ Prove that for any four different peculiar positive integers $A, B, C$ and $D{}$ the following holds:
\[\gcd(A, B, C, D) = 1.\]
2020 EGMO, 5
Consider the triangle $ABC$ with $\angle BCA > 90^{\circ}$. The circumcircle $\Gamma$ of $ABC$ has radius $R$. There is a point $P$ in the interior of the line segment $AB$ such that $PB = PC$ and the length of $PA$ is $R$. The perpendicular bisector of $PB$ intersects $\Gamma$ at the points $D$ and $E$.
Prove $P$ is the incentre of triangle $CDE$.
2015 EGMO, 4
Determine whether there exists an infinite sequence $a_1, a_2, a_3, \dots$ of positive integers
which satisfies the equality \[a_{n+2}=a_{n+1}+\sqrt{a_{n+1}+a_{n}} \] for every positive integer $n$.
2023 EGMO, 3
Let $k$ be a positive integer. Lexi has a dictionary $\mathbb{D}$ consisting of some $k$-letter strings containing only the letters $A$ and $B$. Lexi would like to write either the letter $A$ or the letter $B$ in each cell of a $k \times k$ grid so that each column contains a string from $\mathbb{D}$ when read from top-to-bottom and each row contains a string from $\mathbb{D}$ when read from left-to-right.
What is the smallest integer $m$ such that if $\mathbb{D}$ contains at least $m$ different strings, then Lexi can fill her grid in this manner, no matter what strings are in $\mathbb{D}$?
2015 EGMO, 2
A [i]domino[/i] is a $2 \times 1$ or $1 \times 2$ tile. Determine in how many ways exactly $n^2$ dominoes can be placed without overlapping on a $2n \times 2n$ chessboard so that every $2 \times 2$ square contains at least two uncovered unit squares which lie in the same row or column.
2020 EGMO, 4
A permutation of the integers $1, 2, \ldots, m$ is called [i]fresh[/i] if there exists no positive integer $k < m$ such that the first $k$ numbers in the permutation are $1, 2, \ldots, k$ in some order. Let $f_m$ be the number of fresh permutations of the integers $1, 2, \ldots, m$.
Prove that $f_n \ge n \cdot f_{n - 1}$ for all $n \ge 3$.
[i]For example, if $m = 4$, then the permutation $(3, 1, 4, 2)$ is fresh, whereas the permutation $(2, 3, 1, 4)$ is not.[/i]
2025 EGMO, 1
For a positive integer $N$, let $c_1 < c_2 < \cdots < c_m$ be all positive integers smaller than $N$ that are coprime to $N$. Find all $N \geqslant 3$ such that $$\gcd( N, c_i + c_{i+1}) \neq 1$$ for all $1 \leqslant i \leqslant m-1$
\\[i]Here $\gcd(a, b)$ is the largest positive integer that divides both $a$ and $b$. Integers $a$ and $b$ are coprime if $\gcd(a, b) = 1$.[/i]
\\ [i]Proposed by Paulius Aleknavičius, Lithuania[/i]
2024 EGMO, 2
Let $ABC$ be a triangle with $AC>AB$ , and denote its circumcircle by $\Omega$ and incentre by $I$. Let its incircle meet sides $BC,CA,AB$ at $D,E,F$ respectively. Let $X$ and $Y$ be two points on minor arcs $\widehat{DF}$ and $\widehat{DE}$ of the incircle, respectively, such that $\angle BXD = \angle DYC$. Let line $XY$ meet line $BC$ at $K$. Let $T$ be the point on $\Omega$ such that $KT$ is tangent to $\Omega$ and $T$ is on the same side of line $BC$ as $A$. Prove that lines $TD$ and $AI$ meet on $\Omega$.
[right][i]Tommy Walker Mackay, United Kingdom[/i][/right]
2022 EGMO, 1
Let $ABC$ be an acute-angled triangle in which $BC<AB$ and $BC<CA$. Let point $P$ lie on segment $AB$ and point $Q$ lie on segment $AC$ such that $P \neq B$, $Q \neq C$ and $BQ = BC = CP$. Let $T$ be the circumcenter of triangle $APQ$, $H$ the orthocenter of triangle $ABC$, and $S$ the point of intersection of the lines $BQ$ and $CP$. Prove that $T$, $H$, and $S$ are collinear.
2013 EGMO, 4
Find all positive integers $a$ and $b$ for which there are three consecutive integers at which the polynomial \[ P(n) = \frac{n^5+a}{b} \] takes integer values.
EGMO 2017, 5
Let $n\geq2$ be an integer. An $n$-tuple $(a_1,a_2,\dots,a_n)$ of not necessarily different positive integers is [i]expensive[/i] if there exists a positive integer $k$ such that $$(a_1+a_2)(a_2+a_3)\dots(a_{n-1}+a_n)(a_n+a_1)=2^{2k-1}.$$
a) Find all integers $n\geq2$ for which there exists an expensive $n$-tuple.
b) Prove that for every odd positive integer $m$ there exists an integer $n\geq2$ such that $m$ belongs to an expensive $n$-tuple.
[i]There are exactly $n$ factors in the product on the left hand side.[/i]