A number with $2016$ zeros that is written as $101010 \dots 0101$ is given, in which the zeros and ones alternate. Prove that this number is not prime.

A triangle $ABC$ with area $1$ is given. Anja and Bernd are playing the following game: Anja chooses a point $X$ on side $BC$. Then Bernd chooses a point $Y$ on side $CA$ und at last Anja chooses a point $Z$ on side $AB$. Also, $X,Y$ and $Z$ cannot be a vertex of triangle $ABC$. Anja wants to maximize the area of triangle $XYZ$ and Bernd wants to minimize that area. What is the area of triangle $XYZ$ at the end of the game, if both play optimally?

Let $A,B,C$ and $D$ be points on a circle in this order. The chords $AC$ and $BD$ intersect in point $P$. The perpendicular to $AC$ through C and the perpendicular to $BD$ through $D$ intersect in point $Q$. Prove that the lines $AB$ and $PQ$ are perpendicular.

There are $33$ children in a given class. Each child writes a number on the blackboard, which indicates how many other children possess the same forename as oneself. Afterwards, each child does the same thing with their surname. After they've finished, each of the numbers $0,1,2,\dots,10$ appear at least once on the blackboard. Prove that there are at least two children in this class that have the same forename and surname.

There are $\tfrac{n(n+1)}{2}$ distinct sums of two distinct numbers, if there are $n$ numbers. For which $n \ (n \geq 3)$ do there exist $n$ distinct integers, such that those sums are $\tfrac{n(n-1)}{2}$ consecutive numbers?

Prove that there are infinitely many positive integers that cannot be expressed as the sum of a triangular number and a prime number.

Find all functions $f$ that is defined on all reals but $\tfrac13$ and $- \tfrac13$ and satisfies \[ f \left(\frac{x+1}{1-3x} \right) + f(x) = x \]for all $x \in \mathbb{R} \setminus \{ \pm \tfrac13 \}$.

Each side face of a dodecahedron lies in a uniquely determined plane. Those planes cut the space in a finite number of disjoint regions. Find the number of such regions.