OMCC 2010

Denote by $S(n)$ the sum of the digits of the positive integer $n$ . Find all the solutions of the equation $n(S(n)-1)=2010.$

Let $ABC$ be a triangle and $L$ , $M$ , $N$ be the midpoints of $BC$ , $CA$ and $AB$ , respectively. The tangent to the circumcircle of $ABC$ at $A$ intersects $LM$ and $LN$ at $P$ and $Q$ , respectively. Show that $CP$ is parallel to $BQ$ .

A token is placed in one square of a $m\times n$ board, and is moved according to the following rules:

• In each turn, the token can be moved to a square sharing a side with the one currently occupied.
• The token cannot be placed in a square that has already been occupied.
• Any two consecutive moves cannot have the same direction.

The game ends when the token cannot be moved. Determine the values of $m$ and $n$ for which, by placing the token in some square, all the squares of the board will have been occupied in the end of the game.

Find all positive integers $N$ such that an $N\times N$ board can be tiled using tiles of size $5\times 5$ or $1\times 3$ .

Note: The tiles must completely cover all the board, with no overlappings.

If $p$ , $q$ and $r$ are nonzero rational numbers such that $\sqrt[3]{pq^2}+\sqrt[3]{qr^2}+\sqrt[3]{rp^2}$ is a nonzero rational number, prove that $\frac{1}{\sqrt[3]{pq^2}}+\frac{1}{\sqrt[3]{qr^2}}+\frac{1}{\sqrt[3]{rp^2}}$

is also a rational number.

Let $\Gamma$ and $\Gamma_1$ be two circles internally tangent at $A$ , with centers $O$ and $O_1$ and radii $r$ and $r_1$ , respectively ( $r>r_1$ ). $B$ is a point diametrically opposed to $A$ in $\Gamma$ , and $C$ is a point on $\Gamma$ such that $BC$ is tangent to $\Gamma_1$ at $P$ . Let $A'$ the midpoint of $BC$ . Given that $O_1A'$ is parallel to $AP$ , find the ratio $r/r_1$ .