DE-BUNDESWETTBEWERB 2023

Tick, Trick and Track have 20, 23 and 25 tickets respectively for the carousel at the fair in Duckburg. They agree that they will only ride all three together, for which they must each give up one of their tickets. Also, before a ride, if they want, they can redistribute tickets among themselves as many times as they want according to the following rule: If one has an even number of tickets, he can give half of his tickets to any of the other two. Can it happen that after any trip: (a) exactly one has no ticket left, (b) exactly two have no ticket left, (c) all tickets are given away?

Determine all triples $(x, y, z)$ of integers that satisfy the equation $x^2+ y^2+ z^2 - xy - yz - zx = 3$

Given two parallelograms $ABCD$ and $AECF$ with common diagonal $AC$ , where $E$ and $F$ lie inside parallelogram $ABCD$ . Show: The circumcircles of the triangles $AEB$ , $BFC$ , $CED$ and $DFA$ have one point in common.