CR-MO 2003

Two players $A$ and $B$ participate in the following game. Initially we have a pile of 2003 stones. $A$ plays first, and he picks a divisor of 2003 and removes that number of stones from the pile. Then $B$ picks a divisor of the number of remaining stones, and removes that number of stones from the pile, and so forth. The players who removes the last stone loses. Prove that one of the players has a winning strategy and describe it.

Let $AB$ be a diameter of circle $\omega$ . $\ell$ is the tangent line to $\omega$ at $B$ . Take two points $C$ , $D$ on $\ell$ such that $B$ is between $C$ and $D$ . $E$ , $F$ are the intersections of $\omega$ and $AC$ , $AD$ , respectively, and $G$ , $H$ are the intersections of $\omega$ and $CF$ , $DE$ , respectively. Prove that $AH=AG$ .

If $a>1$ and $b>2$ are positive integers, show that $a^{b}+1 \geq b(a+1)$ , and determine when equality holds.

$S_{1}$ and $S_{2}$ are two circles that intersect at distinct points $P$ and $Q$ . $\ell_{1}$ and $\ell_{2}$ are two parallel lines through $P$ and $Q$ . $\ell_{1}$ intersects $S_{1}$ and $S_{2}$ at points $A_{1}$ and $A_{2}$ , different from $P$ , respectively. $\ell_{2}$ intersects $S_{1}$ and $S_{2}$ at points $B_{1}$ and $B_{2}$ , different from $Q$ , respectively. Show that the perimeters of the triangles $A_{1}QA_{2}$ and $B_{1}PB_{2}$ are equal.

Each of the squares of an $8 \times 8$ board can be colored white or black. Find the number of colorings of the board such that every $2 \times 2$ square contains exactly 2 black squares and 2 white squares.