EGMO 2018

Let $A B C$ be a triangle with $C A=C B$ and $\angle A C B=120^{\circ}$, and let $M$ be the midpoint of $A B$. Let $P$ be a variable point on the circumcircle of $A B C$, and let $Q$ be the point on the segment $C P$ such that $Q P=2 Q C$. It is given that the line through $P$ and perpendicular to $A B$ intersects the line $M Q$ at a unique point $N$.

Prove that there exists a fixed circle such that $N$ lies on this circle for all possible positions of $P$. (Velina Ivanova, Bulgaria)

Consider the set

$$A=\left\{1+\frac{1}{k}: k=1,2,3, \ldots\right\}$$

(a) Prove that every integer $x \geq 2$ can be written as the product of one or more elements of $A$, which are not necessarily different. (b) For every integer $x \geq 2$, let $f(x)$ denote the minimum integer such that $x$ can be written as the product of $f(x)$ elements of $A$, which are not necessarily different. Prove that there exist infinitely many pairs $(x, y)$ of integers with $x \geq 2, y \geq 2$, and

$$f(x y)<f(x)+f(y) .$$

(Pairs $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ are different if $x_{1} \neq x_{2}$ or $\left.y_{1} \neq y_{2}\right)$. (Mihail Baluna, Romania)

The $n$ contestants of an EGMO are named $C_{1}, \ldots, C_{n}$. After the competition they queue in front of the restaurant according to the following rules.

• The Jury chooses the initial order of the contestants in the queue.
• Every minute, the Jury chooses an integer $i$ with $1 \leq i \leq n$.
• If contestant $C_{i}$ has at least $i$ other contestants in front of her, she pays one euro to the Jury and moves forward in the queue by exactly $i$ positions.
• If contestant $C_{i}$ has fewer than $i$ other contestants in front of her, the restaurant opens and the process ends. (a) Prove that the process cannot continue indefinitely, regardless of the Jury's choices. (b) Determine for every $n$ the maximum number of euros that the Jury can collect by cunningly choosing the initial order and the sequence of moves. (Hungary)

A domino is a $1 \times 2$ or $2 \times 1$ tile. Let $n \geq 3$ be an integer. Dominoes are placed on an $n \times n$ board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap.

The value of a row or column is the number of dominoes that cover at least one cell of this row or column. The configuration is called balanced if there exists some $k \geq 1$ such that each row and each column has a value of $k$.

Prove that a balanced configuration exists for every $n \geq 3$, and find the minimum number of dominoes needed in such a configuration. (Merlijn Staps, The Netherlands)

Let $\Gamma$ be the circumcircle of triangle $A B C$. A circle $\Omega$ is tangent to the line segment $A B$ and is tangent to $\Gamma$ at a point lying on the same side of the line $A B$ as $C$. The angle bisector of $\angle B C A$ intersects $\Omega$ at two different points $P$ and $Q$.

Prove that $\angle A B P=\angle Q B C$. (Dominika Regiec, Poland)

(a) Prove that for every real number $t$ such that $0<t<\frac{1}{2}$ there exists a positive integer $n$ with the following property: for every set $S$ of $n$ positive integers there exist two different elements $x$ and $y$ of $S$, and a non-negative integer $m$ (i.e. $m \geq 0$ ), such that

$$|x-m y| \leq t y$$

(b) Determine whether for every real number $t$ such that $0<t<\frac{1}{2}$ there exists an infinite set $S$ of positive integers such that

$$|x-m y|>t y$$

for every pair of different elements $x$ and $y$ of $S$ and every positive integer $m$ (i.e. $m>0$ ). (Merlijn Staps, The Netherlands)