MathNet 2018-part-06 — 181 problems (EN–VI) Question 96

Two players – Andriy and Olesya play the following game. On a table, there is a round cake, which is cut by one of them into $4n$ sectors (pieces) of different weights. The weight of every piece is known to each player. After that they choose pieces for themselves according to the following rules. At first Andriy chooses 1 piece, then Olesya chooses 2 pieces, but so that the pieces that are left after her turn form a sector. Then they take in turns 2 pieces, so that after every step the pieces that are on the table form a sector. Making the last step Andriy takes the last piece. Each player aims that the total weight of the part of the cake he or she took is bigger than the opponent's one. Is it possible that someone surely takes more than half of the whole cake if:
a) Olesya cuts the cake into sectors;
b) Andriy cuts the cake into sectors, but at his first step he is prohibited to take the biggest piece by weight?