In the acute-angled triangle $ABC$ with circumcircle $\omega$ and orthocenter $H$, points $D$ and $E$ are the feet of the perpendiculars from $A$ onto $BC$ and from $B$ onto $AC$ respectively. Let $P$ be a point on the minor arc $BC$ of $\omega$. Points $M$ and $N$ are the feet of the perpendiculars from $P$ onto lines $BC$ and $AC$ respectively. Let $PH$ and $MN$ intersect at $R$. Prove that $\angle DMR=\angle MDR$.
A robot is placed at a point $P$ on the $x$-axis but different from $(0,0)$ and $(1,0)$ and can move only along the axis either to the left or to the right. Two players play the following game. Player $A$ gives a distance and $B$ gives a direction and the robot will move the indicated distance along the indicated direction. Player $A$ aims to move the robot to either $(0,0)$ or $(1,0)$. Player $B$'s aim is to stop $A$ from achieving his aim. For which $P$ can $A$ win?
Let $p\equiv2\pmod3$ be a prime, $k$ a positive integer and let $P(x)=3x^{(2p-1)/3}+3x^{(p+1)/3}+x+1$. For any integer $n$, let $R(n)$ denote the remainder when $n$ is divided by $p$ and let $S=\{0,1,\ldots,p-1\}$. At each step, you can either (a) replace every element $i$ of $S$ with $R(P(i))$ or (b) replace every element $i$ of $S$ with $R(i^k)$. Determine all $k$ such that there exists a finite sequence of steps that reduces $S$ to $\{0\}$.
In an $m\times n$ chessboard $(m,n\ge2)$, some dominoes are placed (without overlap) with each domino covering exactly two adjacent cells. Show that if no more dominoes can be added to the grid, then at least $2/3$ of the chessboard is covered by dominoes.