DE-BUNDESWETTBEWERB 2009

Determine all possible digits $z$ for which $\underbrace{9...9}_{100}z\underbrace{0...0}_{100}9$ is a square number.

Let $P$ be a point inside the triangle $ABC$ and $P_a, P_b ,P_c$ be the symmetric points wrt the midpoints of the sides $BC, CA,AB$ respectively. Prove that that the lines $AP_a, BP_b$ and $CP_c$ are concurrent.

A positive integer is called decimal palindrome if its decimal representation $z_n...z_0$ with $z_n\ne 0$ is mirror symmetric, i.e. if $z_k = z_{n-k}$ applies to all $k= 0, ..., n$ . Show that each integer that is not divisible by $10$ has a positive multiple, which is a decimal palindrome.

At the start of a game there are three boxes with $2008, 2009$ and $2010$ game pieces Anja and Bernd play in turns according to the following rule: When it is your turn, select two boxes, empty them and then distribute the pieces from the third box to the three boxes, such that no box may remain empty.If you can no longer complete a turn, you have lost. Who has a winning strategy when Anja starts?

Let $n$ be an integer that is greater than $1$ . Prove that the following two statements are equivalent: (A) There are positive integers $a, b$ and $c$ that are not greater than $n$ and for which that polynomial $ax^2 + bx + c$ has two different real roots $x_1$ and $x_2$ with $| x_2- x_1 | \le \frac{1}{n}$ (B) The number $n$ has at least two different prime divisors.

Given a triangle $ABC$ and a point $P$ on the side $AB$ . Let $Q$ be the intersection of the straight line $CP$ (different from $C$ ) with the circumcicle of the triangle. Prove the inequality $$\frac{\overline{PQ}}{\overline{CQ}} \le \left(\frac{\overline{AB}}{\overline{AC}+\overline{CB}}\right)^2$$ and that equality holds if and only if the $CP$ is bisector of the angle $ACB$ .

How many diagonals can you draw in a convex $2009$ -gon if in the finished drawing, every drawn diagonal inside the $2009$ -gon may cut at most another drawn diagonal?