MathDB

Let

0<c<1

and

n

a positive integer. Alice and Bob are playing a game. Bob writes

n

integers on the board, not all equal. On a player's turn, they erase two numbers from the board and write their arithmetic mean instead. Alice starts and performs at most

cn

moves. After her, Bob makes moves until there are only two numbers left on the board. Alice wins if these two numbers are different, and otherwise, Bob wins. For which values of

c

does Alice win for all large enough

n

Problem Statement