International Mathematical Olympiad 2010

Determine all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that the equality $$f \big (\lfloor x \rfloor y \big) = f (x) \big \lfloor f (y) \rfloor$$ holds for all $x, y \in \mathbb{R}$. (Here $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$.)

Let $I$ be the incentre of triangle $ABC$ and let $\Gamma$ be its circumcircle. Let the line $AI$ intersect $\Gamma$ again at $D$. Let $E$ be a point on the arc $\widehat{BDC}$ and $F$ a point on the side $BC$ such that $$\angle BAF = \angle CAE < \frac{1}{2}\angle BAC.$$ Finally, let $G$ be the midpoint of the segment $IF$. Prove that the lines $DG$ and $EI$ intersect on $\Gamma$.

Let $\mathbb{N}$ be the set of positive integers. Determine all functions $g\colon \mathbb{N}\to \mathbb{N}$ such that $$\left(g (m) + n\right) \left(m + g (n)\right)$$ is a perfect square for all $m, n \in \mathbb{N}$.

Let $P$ be a point inside the triangle $ABC$. The lines $AP$, $BP$ and $CP$ intersect the circumcircle $\Gamma$ of triangle $ABC$ again at the points $K$, $L$ and $M$ respectively. The tangent to $\Gamma$ at $C$ intersects the line $AB$ at $S$. Suppose that $SC = SP$. Prove that $MK = ML$.

In each of six boxes $B_{1}, B_{2}, B_{3}, B_{4}, B_{5}, B_{6}$ there is initially one coin. There are two types of operation allowed:

Type 1: Choose a nonempty box $B_{j}$ with $1 \leq j \leq 5$. Remove one coin from $B_{j}$ and add two coins to $B_{j+1}$.

Type 2: Choose a nonempty box $B_{k}$ with $1 \leq k \leq 4$. Remove one coin from $B_{k}$ and exchange the contents of (possibly empty) boxes $B_{k+1}$ and $B_{k+2}$.

Determine whether there is a finite sequence of such operations that results in boxes $B_{1}, B_{2}, B_{3}, B_{4}, B_{5}$ being empty and box $B_{6}$ containing exactly $2010^{2010^{2010}}$ coins. (Note that $a^{b^{c}} = a^{(b^{c})}$.)

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive real numbers. Suppose that for some positive integer $s$, we have $$a _ {n} = \max \left\{a _ {k} + a _ {n - k} \mid 1 \leq k \leq n - 1 \right\}$$ for all $n > s$. Prove that there exist positive integers $\ell$ and $N$, with $\ell \leq s$ and such that $a_{n} = a_{\ell} + a_{n - \ell}$ for all $n \geq N$.