Let $\alpha$ and $\beta$ be real numbers with $\beta \ne 0$. Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[f(\alpha f(x)+f(y))=\beta x+f(y)\]holds for all real $x$ and $y$. (Walther Janous)

Let $h$ be a semicircle with diameter $AB$. The two circles $k_1$ and $k_2$, $k_1 \ne k_2$, touch the segment $AB$ at the points $C$ and $D$, respectively, and the semicircle $h$ fom the inside at the points $E$ and $F$, respectively. Prove that the four points $C$, $D$, $E$ and $F$ lie on a circle. (Walther Janous)

Let $n \ge 3$ be an integer. A circle dance is a dance that is performed according to the following rule: On the floor, $n$ points are marked at equal distances along a large circle. At each of these points is a sheet of paper with an arrow pointing either clockwise or counterclockwise. One of the points is labeled "Start". The dancer starts at this point. In each step, he first changes the direction of the arrow at his current position and then moves to the next point in the new direction of the arrow. a) Show that each circle dance visits each point infinitely often. b) How many different circle dances are there? Two circle dances are considered to be the same if they differ only by a finite number of steps at the beginning and then always visit the same points in the same order. (The common sequence of steps may begin at different times in the two dances.) (Birgit Vera Schmidt)

A positive integer is called powerful if all exponents in its prime factorization are $\ge 2$. Prove that there are infinitely many pairs of powerful consecutive positive integers. (Walther Janous)

Determine the smallest real constant $C$ such that the inequality \[(X+Y)^2(X^2+Y^2+C)+(1-XY)^2 \ge 0\]holds for all real numbers $X$ and $Y$. For which values of $X$ and $Y$ does equality hold for this smallest constant $C$? (Walther Janous)

Let $ABC$ be an acute triangle with $AB>AC$. Let $D,E,F$ denote the feet of its altitudes on $BC,AC$ and $AB$, respectively. Let $S$ denote the intersection of lines $EF$ and $BC$. Prove that the circumcircles $k_1$ and $k_2$ of the two triangles $AEF$ and $DES$ touch in $E$. (Karl Czakler)

Initially, the numbers $1, 2, \dots, 2024$ are written on a blackboard. Trixi and Nana play a game, taking alternate turns. Trixi plays first. The player whose turn it is chooses two numbers $a$ and $b$, erases both, and writes their (possibly negative) difference $a-b$ on the blackboard. This is repeated until only one number remains on the blackboard after $2023$ moves. Trixi wins if this number is divisible by $3$, otherwise Nana wins. Which of the two has a winning strategy? (Birgit Vera Schmidt)

Let $ABC$ be an obtuse triangle with orthocenter $H$ and centroid $S$. Let $D$, $E$ and $F$ be the midpoints of segments $BC$, $AC$, $AB$, respectively. Show that the circumcircle of triangle $ABC$, the circumcircle of triangle $DEF$ and the circle with diameter $HS$ have two distinct points in common. (Josef Greilhuber)

Let $n$ be a positive integer and let $z_1,z_2,\dots,z_n$ be positive integers such that for $j=1,2,\dots,n$ the inequalites $z_j \le j$ hold and $z_1+z_2+\dots+z_n$ is even. Prove that the number $0$ occurs among the values \[z_1 \pm z_2 \pm \dots \pm z_n,\]where $+$ or $-$ can be chosen independently for each operation. (Walther Janous)

For each prime number $p$, determine the number of residue classes modulo $p$ which can be represented as $a^2+b^2$ modulo $p$, where $a$ and $b$ are arbitrary integers. (Daniel Holmes)