Five squirrels together have a supply of 2022 nuts. On the first day 2 nuts are added, on the second day 4 nuts, on the third day 6 nuts and so on, i.e. on each further day 2 nuts more are added than on the day before. At the end of any day the squirrels divide the stock among themselves. Is it possible that they all receive the same number of nuts and that no nut is left over?

Eva draws an equilateral triangle and its altitudes. In a first step she draws the center triangle of the equilateral triangle, in a second step the center triangle of this center triangle and so on. After each step Eva counts all triangles whose sides lie completely on drawn lines. What is the minimum number of center triangles she must have drawn so that the figure contains more than 2022 such triangles?

A circle $k$ touches a larger circle $K$ from inside in a point $P$. Let $Q$ be point on $k$ different from $P$. The line tangent to $k$ at $Q$ intersects $K$ in $A$ and $B$. Show that the line $PQ$ bisects $\angle APB$.

For each positive integer $k$ let $a_k$ be the largest divisor of $k$ which is not divisible by $3$. Let $s_n=a_1+a_2+\dots+a_n$. Show that: (a) The number $s_n$ is divisible by $3$ iff the number of ones in the ternary expansion of $n$ is divisible by $3$. (b) There are infinitely many $n$ for which $s_n$ is divisible by $3^3$.

Find all quadrupels $(a, b, c, d)$ of positive real numbers that satisfy the following two equations: \begin{align*} ab + cd &= 8,\\ abcd &= 8 + a + b + c + d. \end{align*}

On a table lie 2022 matches and a regular dice that has the number $a$ on top. Now Max and Moritz play the following game: Alternately, they take away matches according to the following rule, where Max begins: The player to make a move rolls the dice over one of its edges and then takes a way as many matches as the top number shows. The player that cannot make legal move after some number of moves loses. For which $a$ can Moritz force Max to lose?

In an acute triangle $ABC$ with $AC<BC$, lines $m_a$ and $m_b$ are the perpendicular bisectors of sides $BC$ and $AC$, respectively. Further, let $M_c$ be the midpoint of side $AB$. The Median $CM_c$ intersects $m_a$ in point $S_a$ and $m_b$ in point $S_b$; the lines $AS_b$ und $BS_a$ intersect in point $K$. Prove: $\angle ACM_c = \angle KCB$.

Some points in the plane are either colored red or blue. The distance between two points of the opposite color is at most 1. Prove that there exists a circle with diameter $\sqrt{2}$ such that no two points outside of this circle have same color. It is enough to prove this claim for a finite number of colored points.