OMM 2012

Let $\mathcal{C}_1$ be a circumference with center $O$ , $P$ a point on it and $\ell$ the line tangent to $\mathcal{C}_1$ at $P$ . Consider a point $Q$ on $\ell$ different from $P$ , and let $\mathcal{C}_2$ be the circumference passing through $O$ , $P$ and $Q$ . Segment $OQ$ cuts $\mathcal{C}_1$ at $S$ and line $PS$ cuts $\mathcal{C}_2$ at a point $R$ diffferent from $P$ . If $r_1$ and $r_2$ are the radii of $\mathcal{C}_1$ and $\mathcal{C}_2$ respectively, Prove $$\frac{PS}{SR} = \frac{r_1}{r_2}.$$

Let $n \geq 4$ be an even integer. Consider an $n \times n$ grid. Two cells ( $1 \times 1$ squares) are neighbors if they share a side, are in opposite ends of a row, or are in opposite ends of a column. In this way, each cell in the grid has exactly four neighbors. An integer from 1 to 4 is written inside each square according to the following rules:

• If a cell has a 2 written on it, then at least two of its neighbors contain a 1.
• If a cell has a 3 written on it, then at least three of its neighbors contain a 1.
• If a cell has a 4 written on it, then all of its neighbors contain a 1.

Among all arrangements satisfying these conditions, what is the maximum number that can be obtained by adding all of the numbers on the grid?

Prove among any $14$ consecutive positive integers there exist $6$ which are pairwise relatively prime.

The following process is applied to each positive integer: the sum of its digits is subtracted from the number, and the result is divided by $9$ . For example, the result of the process applied to $938$ is $102$ , since $\frac{938-(9 + 3 + 8)}{9} = 102.$ Applying the process twice to $938$ the result is $11$ , applied three times the result is $1$ , and applying it four times the result is $0$ . When the process is applied one or more times to an integer $n$ , the result is eventually $0$ . The number obtained before obtaining $0$ is called the house of $n$ .

How many integers less than $26000$ share the same house as $2012$ ?

Some frogs, some red and some others green, are going to move in an $11 \times 11$ grid, according to the following rules. If a frog is located, say, on the square marked with

Consider an acute triangle $ABC$ with circumcircle $\mathcal{C}$ . Let $H$ be the orthocenter of $ABC$ and $M$ the midpoint of $BC$ . Lines $AH$ , $BH$ and $CH$ cut $\mathcal{C}$ again at points $D$ , $E$ , and $F$ respectively; line $MH$ cuts $\mathcal{C}$ at $J$ such that $H$ lies between $J$ and $M$ . Let $K$ and $L$ be the incenters of triangles $DEJ$ and $DFJ$ respectively. Prove $KL$ is parallel to $BC$ .