# [Solutions] Middle European Mathematical Olympiad 2016

### Individual Competition

1. Let $n \ge 2$ be an integer, and let $x_1, x_2, \ldots, x_n$ be reals for which
• $x_j > -1$ for $j = 1, 2, \ldots, n$ and
• $x_1 + x_2 + \ldots + x_n = n.$
Prove that $$\sum_{j = 1}^{n} \frac{1}{1 + x_j} \ge \sum_{j = 1}^{n} \frac{x_j}{1 + x_j^2}$$and determine when does the equality occur.
2. There are $n \ge 3$ positive integers written on a board. A move consists of choosing three numbers $a, b, c$ written from the board such that there exists a non-degenerate non-equilateral triangle with sides $a, b, c$ and replacing those numbers with $a + b - c, b + c - a$ and $c + a - b$. Prove that a sequence of moves cannot be infinite.
3. Let $ABC$ be an acute triangle such that $\angle BAC > 45^{\circ}$ with circumcenter $O$. A point $P$ is chosen inside triangle $ABC$ such that $A, P, O, B$ are concyclic and the line $BP$ is perpendicular to the line $CP$. A point $Q$ lies on the segment $BP$ such that the line $AQ$ is parallel to the line $PO$. Prove that $\angle QCB = \angle PCO$.
4. Find all $f : \mathbb{N} \to \mathbb{N}$ such that $f(a) + f(b)$ divides $2(a + b - 1)$ for all $a, b \in \mathbb{N}$.
Remark: $\mathbb{N} = \{ 1, 2, 3, \ldots \}$ denotes the set of the positive integers.

### Team Competition

1. Find all triples $(a, b, c)$ of real numbers such that $$a^2 + ab + c = 0,$$ $$b^2 + bc + a = 0,$$ $$c^2 + ca + b = 0.$$
2. Let $\mathbb{R}$ denote the set of the reals. Find all $f : \mathbb{R} \to \mathbb{R}$ such that $$f(x)f(y) = xf(f(y-x)) + xf(2x) + f(x^2)$$for all real $x, y$.
3. A $8 \times 8$ board is given, with sides directed north-south and east-west. It is divided into $1 \times 1$ cells in the usual manner. In each cell, there is most one house. A house occupies only one cell. A house is in the shade if there is a house in each of the cells in the south, east and west sides of its cell. In particular, no house placed on the south, east or west side of the board is in the shade. Find the maximal number of houses that can be placed on the board such that no house is in the shade.
4. An exam was taken by some students. Each problem was worth $1$ point for the correct answer, and $0$ points for an incorrect one. For each question, at least one student answered it correctly. Also, there are two students with different scores on the exam. Prove that there exists a question for which the following holds: The average score of the students who answered the question correctly is greater than the average score of the students who didn't.
5. Let $ABC$ be an acute triangle for which $AB \neq AC$, and let $O$ be its circumcenter. Line $AO$ meets the circumcircle of $ABC$ again in $D$, and the line $BC$ in $E$. The circumcircle of $CDE$ meets the line $CA$ again in $P$. The lines $PE$ and $AB$ intersect in $Q$. Line passing through $O$ parallel to the line $PE$ intersects the $A$-altitude of $ABC$ in $F$. Prove that $FP = FQ$.
6. Let $ABC$ be a triangle for which $AB \neq AC$. Points $K$, $L$, $M$ are the midpoints of the sides $BC$, $CA$, $AB$. The incircle of $ABC$ with center $I$ is tangent to $BC$ in $D$. A line passing through the midpoint of $ID$ perpendicular to $IK$ meets the line $LM$ in $P$. Prove that $\angle PIA = 90 ^{\circ}$.
7. A positive integer $n$ is Mozart if the decimal representation of the sequence $1, 2, \ldots, n$ contains each digit an even number of times. Prove that:
a) All Mozart numbers are even.
b) There are infinitely many Mozart numbers.
8. For a positive integer $n$, the equation $a^2 + b^2 + c^2 + n = abc$ is given in the positive integers. Prove that
a) There does not exist a solution $(a, b, c)$ for $n = 2017$.
b) For $n = 2016$, $a$ is divisible by $3$ for all solutions $(a, b, c)$.
c) There are infinitely many solutions $(a, b, c)$ for $n = 2016$.
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