OMCC 2009

Let $P$ be the product of all non-zero digits of the positive integer $n$ . For example, $P(4) = 4$ , $P(50) = 5$ , $P(123) = 6$ , $P(2009) = 18$ . Find the value of the sum: P(1) + P(2) + ... + P(2008) + P(2009).

\item Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $B$ . Consider a circle $\Gamma$ contained in $\Gamma_1$ and $\Gamma_2$ , which is tangent to both of them at $D$ and $E$ respectively. Let $C$ be one of the intersection points of line $AB$ with $\Gamma$ , $F$ be the intersection of line $EC$ with $\Gamma_2$ and $G$ be the intersection of line $DC$ with $\Gamma_1$ . Let $H$ and $I$ be the intersection points of line $ED$ with $\Gamma_1$ and $\Gamma_2$ respectively. Prove that $F$ , $G$ , $H$ and $I$ are on the same circle.

There are 2009 boxes numbered from 1 to 2009, some of which contain stones. Two players, $A$ and $B$ , play alternately, starting with $A$ . A move consists in selecting a non-empty box $i$ , taking one or more stones from that box and putting them in box $i + 1$ . If $i = 2009$ , the selected stones are eliminated. The player who removes the last stone wins a) If there are 2009 stones in the box 2 and the others are empty, find a winning strategy for either player. b) If there is exactly one stone in each box, find a winning strategy for either player.

We wish to place natural numbers around a circle such that the following property is satisfied: the absolute values of the differences of each pair of neighboring numbers are all different. a) Is it possible to place the numbers from 1 to 2009 satisfying this property b) Is it possible to suppress one of the numbers from 1 to 2009 in such a way that the remaining 2008 numbers can be placed satisfying the property

Given an acute and scalene triangle $ABC$ , let $H$ be its orthocenter, $O$ its circumcenter, $E$ and $F$ the feet of the altitudes drawn from $B$ and $C$ , respectively. Line $AO$ intersects the circumcircle of the triangle again at point $G$ and segments $FE$ and $BC$ at points $X$ and $Y$ respectively. Let $Z$ be the point of intersection of line $AH$ and the tangent line to the circumcircle at $G$ . Prove that $HX$ is parallel to $YZ$ .

Find all prime numbers $p$ and $q$ such that $p^3 - q^5 = (p + q)^2$ .