OMM 2020 Problem 3

Let $n\ge 3$ be an integer. Two players, Ana and Beto, play the following game. Ana tags the vertices of a regular $n$ - gon with the numbers from $1$ to $n$ , in any order she wants. Every vertex must be tagged with a different number. Then, we place a turkey in each of the $n$ vertices. These turkeys are trained for the following. If Beto whistles, each turkey moves to the adjacent vertex with greater tag. If Beto claps, each turkey moves to the adjacent vertex with lower tag. Beto wins if, after some number of whistles and claps, he gets to move all the turkeys to the same vertex. Ana wins if she can tag the vertices so that Beto can't do this. For each $n\ge 3$ , determine which player has a winning strategy.

Proposed by Victor and Isaías de la Fuente