IZhO 2006

Solve in positive integers the equation $$n = \varphi(n) + 402 ,$$ where $\varphi(n)$ is the number of positive integers less than $n$ having no common prime factors with $n$ .

Let $ABC$ be a triangle and $K$ and $L$ be two points on $(AB)$ , $(AC)$ such that $BK = CL$ and let $P = CK\cap BL$ . Let the parallel through $P$ to the interior angle bisector of $\angle BAC$ intersect $AC$ in $M$ . Prove that $CM = AB$ .

Let $m\geq n\geq 4$ be two integers. We call a $m\times n$ board filled with 0's or 1's good if

1. not all the numbers on the board are 0 or 1;

2. the sum of all the numbers in $3\times 3$ sub-boards is the same;

3. the sum of all the numbers in $4\times 4$ sub-boards is the same.

Find all $m,n$ such that there exists a good $m\times n$ board.

In a pile you have 100 stones. A partition of the pile in $k$ piles is good if:

1. the small piles have different numbers of stones;

2. for any partition of one of the small piles in 2 smaller piles, among the $k + 1$ piles you get 2 with the same number of stones (any pile has at least 1 stone).

Find the maximum and minimal values of $k$ for which this is possible.

Let $a,b,c,d$ be real numbers with sum 0. Prove the inequality: $$(ab + ac + ad + bc + bd + cd)^2 + 12\geq 6(abc + abd + acd + bcd).$$

Let $ABCDEF$ be a convex hexagon such that $AD = BC + EF$ , $BE = AF + CD$ , $CF = DE + AB$ . Prove that: $$\frac {AB}{DE} = \frac {CD}{AF} = \frac {EF}{BC}.$$