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[Solutions] European Girls Mathematical Olympiad 2018

  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 $P$. 
  2. Consider the set \[A = \left\{1+\frac{1}{k} : k=1,2,3,4,\cdots \right\}.\] 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 \[f(xy)<f(x)+f(y).\] Note. Pairs $(x_1,y_1)$ and $(x_2,y_2)$ are different if $x_1 \neq x_2$ or $y_1 \neq y_2$.
  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.
    • 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 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.
    1. 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 dominocovers 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. 
    2. 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 $\angle ABP = \angle QBC$. 
    3. 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. $m > 0$).

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    MOlympiad: [Solutions] European Girls Mathematical Olympiad 2018
    [Solutions] European Girls Mathematical Olympiad 2018
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