DE-BUNDESWETTBEWERB 2024

Let $ABCD$ be a parallelogram whose diagonals intersect in $M$ . Suppose that the circumcircle of $ABM$ intersects the segment $AD$ in a point $E \ne A$ and that the circumcircle of $EMD$ intersects the segment $BE$ in a point $F \ne E$ . Show that $\angle ACB=\angle DCF$ .

For positive integers $p$ , $q$ and $r$ we are given $p \cdot q \cdot r$ unit cubes. We drill a hole along the space diagonal of each of these cubes and then tie them to a very thin thread of length $p \cdot q \cdot r \cdot \sqrt{3}$ like a string of pearls.

We now want to construct a cuboid of side lengths $p$ , $q$ and $r$ out of the cubes, without tearing the thread. a) For which numbers $p$ , $q$ and $r$ is this possible? b) For which numbers $p$ , $q$ and $r$ is this possible in a way such that both ends of the thread coincide?

Determine all pairs $(x,y)$ of integers satisfying $$(x+2)^4-x^4=y^3.$$

Determine the set of all real numbers $r$ for which there exists an infinite sequence $a_1,a_2,\dots$ of positive integers satisfying the following three properties: (1) No number occurs more than once in the sequence. (2) The sum of two different elements of the sequence is never a power of two. (3) For all positive integers $n$ , we have $a_n<r \cdot n$ .