OBM 2009

Let $q = 2p+1$ , $p, q > 0$ primes. Prove that there exists a multiple of $q$ whose digits sum in decimal base is positive and at most $3$ .

There are $2009$ pebbles in some points $(x,y)$ with both coordinates integer. A operation consists in choosing a point $(a,b)$ with four or more pebbles, removing four pebbles from $(a,b)$ and putting one pebble in each of the points $$(a,b-1),\ (a,b+1),\ (a-1,b),\ (a+1,b)$$

Show that after a finite number of operations each point will necessarily have at most three pebbles. Prove that the final configuration doesn't depend on the order of the operations.

Prove that there exists a positive integer $n_0$ with the following property: for each integer $n \geq n_0$ it is possible to partition a cube into $n$ smaller cubes.

Let $ABC$ be a triangle and $O$ its circumcenter. Lines $AB$ and $AC$ meet the circumcircle of $OBC$ again in $B_1\neq B$ and $C_1 \neq C$ , respectively, lines $BA$ and $BC$ meet the circumcircle of $OAC$ again in $A_2\neq A$ and $C_2\neq C$ , respectively, and lines $CA$ and $CB$ meet the circumcircle of $OAB$ in $A_3\neq A$ and $B_3\neq B$ , respectively. Prove that lines $A_2A_3$ , $B_1B_3$ and $C_1C_2$ have a common point.

Let $n > 3$ be a fixed integer and $x_1,x_2,\ldots, x_n$ be positive real numbers. Find, in terms of $n$ , all possible real values of $${x_1\over x_n+x_1+x_2} + {x_2\over x_1+x_2+x_3} + {x_3\over x_2+x_3+x_4} + \cdots + {x_{n-1}\over x_{n-2}+x_{n-1}+x_n} + {x_n\over x_{n-1}+x_n+x_1}$$