2018 EGMO Problems

Day 1

Problem 1

Let $ABC$ be a triangle with $CA=CB$ and $\angle{ACB}=120^\circ$ , and let $M$ be the midpoint of $AB$ . Let $P$ be a variable point of the circumcircle of $ABC$ , and let $Q$ be the point on the segment $CP$ such that $QP = 2QC$ . It is given that the line through $P$ and perpendicular to $AB$ intersects the line $MQ$ at a unique point $N$ . Prove that there exists a fixed circle such that $N$ lies on this circle for all possible positions of $P$ .

Solution

Problem 2

Consider the set $A = \left\{1+\frac{1}{k} : k=1,2,3,4,\cdots \right\}.$

[list=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.

[*]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 $f(xy)<f(x)+f(y).$ (Pairs $(x_1,y_1)$ and $(x_2,y_2)$ are different if $x_1 \neq x_2$ or $y_1 \neq y_2$ ). [/list]

Solution

Problem 3

The $n$ contestant of EGMO are named $C_1, C_2, \cdots C_n$ . After the competition, they queue in front of the restaurant according to the following rules. [list] [*]The Jury chooses the initial order of the contestants in the queue. [*]Every minute, the Jury chooses an integer $i$ with $1 \leq i \leq n$ . [list] [*]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 process ends. [/list] [/list] [list=a] [*]Prove that the process cannot continue indefinitely, regardless of the Jury’s choices. [*]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. [/list]

Solution

Day 2

Problem 4

A domino is a $1 \times 2$ or $2 \times 1$ tile. Let $n \ge 3$ 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 $k \ge 1$ such that each row and each column has a value of $k$ . Prove that a balanced configuration exists for every $n \ge 3$ , and find the minimum number of dominoes needed in such a configuration.

Solution

Problem 5

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

Solution

Problem 6

[list=a] [*]Prove that for every real number $t$ such that $0 < t < \tfrac{1}{2}$ 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. $m \ge 0$ ), such that $|x-my|\leq ty.$ [*]Determine whether for every real number $t$ such that $0 < t < \tfrac{1}{2}$ there exists an infinite set $S$ of positive integers such that $|x-my| > ty$ for every pair of different elements $x$ and $y$ of $S$ and every positive integer $m$ (i.e. $m > 0$ ).

Solution

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2018 EGMO Problems

Contents

Day 1

Problem 1

Problem 2

Problem 3

Day 2

Problem 4

Problem 5

Problem 6