UA-UMO 2018 Problem 4

Two players - Andriy and Olesya play the following game. On a table there is a round cake, which is cut by one of the players into $2n$ ( $n>1$ ) sectors (pieces), pairwise distinct in weights. Weight of each piece is known to both players. Then they take pieces according to the following rules. At first, Olesya takes $1$ piece, then Andriy takes $2$ pieces such that the remaining pieces form a sector. Then they in turns take $2$ pieces such that after each turn the remaining pieces form a sector. In their last move, one of the players takes one last piece. Each of the players aims to get more of a cake than the opponent. For which $n$ can Olesya cut the cake such that she wins if in her first move she takes the smallest piece?

Proposed by Bohdan Rublyov