Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$, $$
f(\left\lfloor x\right\rfloor y) = f(x)\left\lfloor f(y)\right\rfloor.
$$ Here $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z$.
Let $I$ be the incenter of a triangle $ABC$ and let $\Gamma$ be its circumcircle. Let line $AI$ intersect $\Gamma$ again at $D$. Let $E$ be a point on arc $\widehat{BDC}$ and $F$ a point on side $BC$ such that $$
\angle BAF = \angle CAE \lt \tfrac12 \angle BAC.
$$ Finally, let $G$ be the midpoint of $\overline{IF}$. Prove that $\overline{DG}$ and $\overline{EI}$ intersect on $\Gamma$.
Find all functions $g \colon \mathbb{Z}_{\gt 0} \to \mathbb{Z}_{\gt 0}$ such that $$
\left( g(m)+n \right)\left( g(n)+m \right)
$$ is a perfect square for every pair of positive integers $m,n$.
Let $P$ be a point interior to triangle $ABC$ (with $CA \neq CB$). The lines $AP$, $BP$ and $CP$ meet again its circumcircle $\Gamma$ at $K$, $L$, $M$, respectively. The tangent line at $C$ to $\Gamma$ meets the line $AB$ at $S$. Show that from $SC = SP$ follows $MK = ML$.
Each of the six boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, $B_6$ initially contains one coin. The following two types of operations are allowed:
1. Choose a non-empty box $B_j$, $1\leq j \leq 5$, remove one coin from $B_j$ and add two coins to $B_{j+1}$;
2. Choose a non-empty box $B_k$, $1\leq k \leq 4$, remove one coin from $B_k$ and swap the contents (possibly empty) of the boxes $B_{k+1}$ and $B_{k+2}$.
Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$ become empty, while box $B_6$ contains exactly $2010^{2010^{2010}}$ coins. Note that $a^{b^c}=a^{(b^c)}$.
Let $a_1, a_2, a_3, \dots$ be a sequence of positive real numbers, and $s$ be a positive integer, such that $$
a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \text{ for all } n \gt s.
$$ Prove there exist positive integers $\ell \leq s$ and $N$, such that $$
a_n = a_{\ell} + a_{n - \ell} \text{ for all } n \ge N.
$$