BALKAN 2013

We denote the angles of the triangle by $\alpha, \beta$ and $\gamma$ as usual. Since $\angle K M P=90^{\circ}-\frac{\beta}{2}$, it suffices to prove that $\angle K L P=90^{\circ}+\frac{\beta}{2}$, which is equivalent to $\angle K L C=\frac{\beta}{2}$. Let $I$ be the incenter of triangle $A B C$ and let $D$ be the tangency point of the incircle with $A B$. Since $C K \| I B$ and $C L \| I A$, it holds that $\angle K C L=\angle A I B$. Moreover, from $C N=A D=\frac{b+c-a}{2}$ and $\angle K C N=\frac{\beta}{2}$ we obtain $C K=C N \cos \frac{\beta}{2}=$ $A D \cos \frac{\beta}{2}=A I \cos \frac{\alpha}{2} \cos \frac{\beta}{2}$ and analogously $C L=B I \cos \frac{\alpha}{2} \cos \frac{\beta}{2}$, which imply $\frac{C K}{C L}=\frac{A I}{B I}$. Hence the triangles $K C L$ and $A I B$ are similar, and thus $\angle K L C=$ $\angle A B I=\frac{\beta}{2}$.

Determine all positive integers $x, y$ and $z$ such that

$$x^{5}+4^{y}=2013^{z}$$

Let $S$ be the set of positive real numbers. Find all functions $f: S^{3} \rightarrow S$ such that, for all positive real numbers $x, y, z$ and $k$, the following three conditions are satisfied: (i) $x f(x, y, z)=z f(z, y, x)$; (ii) $f\left(x, k y, k^{2} z\right)=k f(x, y, z)$; (iii) $f(1, k, k+1)=k+1$. (United Kingdom)

In a mathematical competition some competitors are friends; friendship is always mutual, that is to say that when $A$ is a friend of $B$, then also $B$ is a friend of $A$. We say that $n \geq 3$ different competitors $A_{1}, A_{2}, \ldots, A_{n}$ form a weakly-friendly cycle if $A_{i}$ is not a friend of $A_{i+1}$ for $1 \leq i \leq n\left(A_{n+1}=A_{1}\right)$, and there are no other pairs of non-friends among the components of this cycle. The following property is satisfied: for every competitor $C$, and every weakly-friendly cycle $\mathscr{S}$ of competitors not including $C$, the set of competitors $D$ in $\mathscr{S}$ which are not friends of $C$ has at most one element.

Prove that all competitors of this mathematical competition can be arranged into three rooms, such that every two competitors that are in the same room are friends. (Serbia) Time allowed: 270 minutes. Each problem is worth 10 points.