Austrian Mathematical Olympiad

Anna's strategy consists of always adding a single match to the bowl while there are less than 1350 matches inside. With this strategy, she will always change an even number of matches to an odd number of matches, so that Bernd is forced to choose an odd divisor and give her an even number of matches again. Bernd will have at most 1350 matches and can add at most a third, since there is no larger odd proper divisor. Since $1350+\frac{1}{3}\cdot 1350=1350+450=1800<2024$, he cannot reach more than 2024 matches in this phase of the game. Therefore, there will come a turn where Anna starts with an even number of at least 1350 matches. She can add half of them and obtains at least $1350+\frac{1}{2}\cdot 1350=1350+675=2025$ matches and has won.