Polish Mathematical Olympiad 2007

  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
  3. 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.
  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 $n\geq 1$.

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