Question 101
Two players – Andriy and Olesya play the following game. On a table there is a rounded cake, which is cut by one of them into $2n$, $n \gt 1$ different in weight sectors (pieces). Weight of every piece is known by each player. After that they choose pieces according the following rules. At first Olesya chooses 1 piece, then Andriy chooses 2 pieces, but so, that pieces, that are left on table after his turn, form a sector. Then they choose in turns 2 pieces each so, that after every step pieces, that are left on the table, form a sector. By his last turn one of players takes the last piece. Each of players aims, that total weight of part of the cake, he or she took, is bigger, than opponent's one. For which $n$ Olesya can cut the cake so, that if by her first step she takes the smallest piece, she wins?
(Bogdan Rublyov)
(Bogdan Rublyov)