# [Solutions] Middle European Mathematical Olympiad 2010

### Individual Competition

1. Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x\mbox{,}y\in\mathbb{R}$, we have $f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y)\mbox{.}$
2. All positive divisors of a positive integer $N$ are written on a blackboard. Two players $A$ and $B$ play the following game taking alternate moves. In the firt move, the player $A$ erases $N$. If the last erased number is $d$, then the next player erases either a divisor of $d$ or a multiple of $d$. The player who cannot make a move loses. Determine all numbers $N$ for which $A$ can win independently of the moves of $B$.
3. We are given a cyclic quadrilateral $ABCD$ with a point $E$ on the diagonal $AC$ such that $AD=AE$ and $CB=CE$. Let $M$ be the center of the circumcircle $k$ of the triangle $BDE$. The circle $k$ intersects the line $AC$ in the points $E$ and $F$. Prove that the lines $FM$, $AD$ and $BC$ meet at one point.
4. Find all positive integers $n$ which satisfy the following tow conditions
• $n$ has at least four different positive divisors;
• for any divisors $a$ and $b$ of $n$ satisfying $1<a<b<n$, the number $b-a$ divides $n$.

### Team Competition

1. Three strictly increasing sequences $a_1, a_2, a_3, \ldots,\qquad b_1, b_2, b_3, \ldots,\qquad c_1, c_2, c_3, \ldots$ of positive integers are given. Every positive integer belongs to exactly one of the three sequences. For every positive integer $n$, the following conditions hold: $c_{a_n}=b_n+1$; $a_{n+1}>b_n$; the number $c_{n+1}c_{n}-(n+1)c_{n+1}-nc_n$ is even. Find $a_{2010}$, $b_{2010}$ and $c_{2010}$.
2. For each integer $n\geqslant2$, determine the largest real constant $C_n$ such that for all positive real numbers $a_1, \ldots, a_n$ we have $\frac{a_1^2+\ldots+a_n^2}{n}\geqslant\left(\frac{a_1+\ldots+a_n}{n}\right)^2+C_n\cdot(a_1-a_n)^2\mbox{.}$
3. In each vertex of a regular $n$-gon, there is a fortress. At the same moment, each fortress shoots one of the two nearest fortresses and hits it. The result of the shooting is the set of the hit fortresses; we do not distinguish whether a fortress was hit once or twice. Let $P(n)$ be the number of possible results of the shooting. Prove that for every positive integer $k\geqslant 3$, $P(k)$ and $P(k+1)$ are relatively prime.
4. Let $n$ be a positive integer. A square $ABCD$ is partitioned into $n^2$ unit squares. Each of them is divided into two triangles by the diagonal parallel to $BD$. Some of the vertices of the unit squares are colored red in such a way that each of these $2n^2$ triangles contains at least one red vertex. Find the least number of red vertices.
5. The incircle of the triangle $ABC$ touches the sides $BC$, $CA$, and $AB$ in the points $D$, $E$ and $F$, respectively. Let $K$ be the point symmetric to $D$ with respect to the incenter. The lines $DE$ and $FK$ intersect at $S$. Prove that $AS$ is parallel to $BC$.
6. Let $A$, $B$, $C$, $D$, $E$ be points such that $ABCD$ is a cyclic quadrilateral and $ABDE$ is a parallelogram. The diagonals $AC$ and $BD$ intersect at $S$ and the rays $AB$ and $DC$ intersect at $F$. Prove that $\sphericalangle{AFS}=\sphericalangle{ECD}$.
7. For a nonnegative integer $n$, define $a_n$ to be the positive integer with decimal representation $1\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}1\mbox{.}$ Prove that $\frac{a_n}{3}$ is always the sum of two positive perfect cubes but never the sum of two perfect squares.
8. We are given a positive integer $n$ which is not a power of two. Show that ther exists a positive integer $m$ with the following two properties:
• $m$ is the product of two consecutive positive integers;
• the decimal representation of $m$ consists of two identical blocks with $n$ digits.
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