1. 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 $$\frac{LM}{MK}=\frac{AD}{DB}$$
2. 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
4. 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 $$n^{2}\leq k \leq m \leq (n+1)^{2}.$$
5. In tetrahedron $ABCD$ following equalities hold \begin{align}\angle BAC+\angle BDC&=\angle ABD+\angle ACD, \\ \angle BAD+\angle BCD&=\angle ABC+\angle ADC.\end{align} Prove that center of sphere circumscribed about ABCD lies on a line through midpoints of $AB$ and $CD$.
6.  Sequence $a_{0}, a_{1}, a_{2},...$ is determined by $a_{0}=-1$ and $$a_{n}+\frac{a_{n-1}}{2}+\frac{a_{n-2}}{3}+...+\frac{a_{1}}{n}+\frac{a_{0}}{n+1}=0,\,\forall n\geq 1.$$ Prove that $a_{n}>0$ for $n\geq 1$.