PL-OM 2007

In acute triangle $ABC$ point $O$ is circumcenter, segment $CD$ is a height, point $E$ lies on side $AB$ and point $M$ is a midpoint of $CE$ . Line through $M$ perpendicular to $OM$ cuts lines $AC$ and $BC$ respectively in $K$ , $L$ . Prove that $\frac{LM}{MK}=\frac{AD}{DB}$

Positive integer will be called white, if it is equal to $1$ or is a product of even number of primes (not necessarily distinct). Rest of the positive integers will be called black. Determine whether there exists a positive integer which sum of white divisors is equal to sum of black divisors

Plane is divided with horizontal and vertical lines into unit squares. Into each square we write a positive integer so that each positive integer appears exactly once. Determine whether it is possible to write numbers in such a way, that each written number is a divisor of a sum of its four neighbours.

Given is an integer $n\geq 1$ . Find out the number of possible values of products $k \cdot m$ , where $k,m$ are integers satisfying $n^{2}\leq k \leq m \leq (n+1)^{2}$ .

In tetrahedron $ABCD$ following equalities hold: $\angle BAC+\angle BDC=\angle ABD+\angle ACD$ $\angle BAD+\angle BCD=\angle ABC+\angle ADC$ Prove that center of sphere circumscribed about ABCD lies on a line through midpoints of $AB$ and $CD$ .

Sequence $a_{0}, a_{1}, a_{2},...$ is determined by $a_{0}=-1$ and $a_{n}+\frac{a_{n-1}}{2}+\frac{a_{n-2}}{3}+...+\frac{a_{1}}{n}+\frac{a_{0}}{n+1}=0$ for $n\geq 1$ Prove that $a_{n}>0$ for $n\geq 1$