2007 Polish Mathematical Olympiad Third Round

Day 1

Problem 1

In an acute triangle $ABC$ let $O$ be the center of the circumcircle, segment $CD$ be the height, point $E$ be any point on the segment $AB$ and point $M$ be the middle of segment $CE$. The line perpendicular to $OM$ and passing through $M$ passes the lines $AC, BC$ accordingly in points $K,L$. Prove that \[\frac{LM}{MK}=\frac{AD}{DB}\].

Problem 2

We call a positive integer $\textit{white}$ if it equals 1 or is the product of an even number of prime numbers (not necessarily distinct). All the other positive integers will be called $\textit{black}$. Examine if there exists any positive integer such that sum of its white divisors equals the sum of its black divisors.

Problem 3

The plane has been divided with horizontal and vertical lines into unit squares. In each square we write one positive integer so that every positive integer appears only once in the plane. Is it possible to write the numbers in such a way that every number is the divisor of the sum of numbers in 4 neighbour squares.?

Day 2

Problem 4

Let $n \geq 1$ be an integer. Determine the number of possible values of the product $km$, where $k,m$ are integers satisfying the inequalities \[n^2 \leq k \leq m \leq (n+1)^2\].

Problem 5

Tetrahedron $ABCD$ satisfies the equalities \[\angle BAC + \angle BDC = \angle ABD + \angle ACD,\] \[\angle BAD + \angle BCD = \angle ABC + \angle ADC.\] Prove that the center of sphere circumscribed on $ABCD$ lies on the line passing through the centres of edges $AB$ and $CD$.

Problem 6

Sequence $a_0, a_1, a_2, \ldots$ is described by conditions: $a_0 = -1$ and \[a_n + \frac{a_{n-1}}{2} + \frac{a_{n-2}}{3} + \ldots + \frac{a_1}{n} + \frac{a_0}{n+1}=0 \qquad \text{for } n \geq 1.\] Prove that $a_n > 0$ for $n \geq 1$.