2009 Mediterranean Mathematics Olympiad

1

Determine all integers $n\ge1$ for which there exists $n$ real numbers $x_1,\ldots,x_n$ in the closed interval $[-4,2]$ such that the following three conditions are fulfilled: - the sum of these real numbers is at least $n$. - the sum of their squares is at most $4n$. - the sum of their fourth powers is at least $34n$. (Proposed by Gerhard Woeginger, Austria)

2

Let $ABC$ be a triangle with $90^\circ \ne \angle A \ne 135^\circ$. Let $D$ and $E$ be external points to the triangle $ABC$ such that $DAB$ and $EAC$ are isoscele triangles with right angles at $D$ and $E$. Let $F = BE \cap CD$, and let $M$ and $N$ be the midpoints of $BC$ and $DE$, respectively. Prove that, if three of the points $A$, $F$, $M$, $N$ are collinear, then all four are collinear.

3

Decide whether the integers $1,2,\ldots,100$ can be arranged in the cells $C(i, j)$ of a $10\times10$ matrix (where $1\le i,j\le 10$), such that the following conditions are fullfiled: i) In every row, the entries add up to the same sum $S$. ii) In every column, the entries also add up to this sum $S$. iii) For every $k = 1, 2, \ldots, 10$ the ten entries $C(i, j)$ with $i-j\equiv k\bmod{10}$ add up to $S$. (Proposed by Gerhard Woeginger, Austria)

4

Let $x,y,z$ be positive real numbers. Prove that \[ \sum_{cyclic} \frac{xy}{xy+x^2+y^2} ~\le~ \sum_{cyclic} \frac{x}{2x+z} \] (Proposed by Šefket Arslanagić, Bosnia and Herzegovina)