DE-BUNDESWETTBEWERB 2010

There are $9999$ rods with lengths $1, 2, ..., 9998, 9999$ . The players Anja and Bernd alternately remove one of the sticks, with Anja starting. The game ends when there are only three bars left. If from those three bars, a not degenerate triangle can be constructed then Anja wins, otherwise Bernd. Who has a winning strategy?

On the sides of a triangle $XYZ$ to the outside construct similar triangles $YDZ, EXZ ,YXF$ with circumcenters $K, L ,M$ respectively. Here are $\angle ZDY = \angle ZXE = \angle FXY$ and $\angle YZD = \angle EZX = \angle YFX$ . Show that the triangle $KLM$ is similar to the triangles .

Find all numbers that can be expressed in exactly $2010$ different ways as the sum of powers of two with non-negative exponents, each power appearing as a summand at most three times. A sum can also be made from just one summand.

Let $a, b, c$ be the side lengths of an non-degenerate triangle with $a \le b \le c$ . With $t (a, b, c)$ denote the minimum of the quotients $\frac{b}{a}$ and $\frac{c}{b}$ . Find all values that $t (a, b, c)$ can take.

The sequence of numbers $a_1, a_2, a_3, ...$ is defined recursively by $a_1 = 1, a_{n + 1} = \lfloor \sqrt{a_1+a_2+...+a_n} \rfloor$ for $n \ge 1$ . Find all numbers that appear more than twice at this sequence.

Given an acute-angled triangle $ABC$ . Let $CB$ be the altitude and $E$ a random point on the line $CD$ . Finally, let $P, Q, R$ and $S$ are the projections of $D$ on the straight lines $AC, AE, BE$ and $BC$ . Prove that the points $P, Q, R$ and $S$ lie either on a circle or on one straight line.

In the following, let $N_0$ denotes the set of non-negative integers. Find all polynomials $P(x)$ that fulfill the following two properties: (1) All coefficients of $P(x)$ are from $N_0$ . (2) Exists a function $f : N_0 \to N_0$ such as $f (f (f (n))) = P (n)$ for all $n \in N_0$ .