KöMaL 2014 A615

Basil and Peter play the following game. Basil writes 100 real numbers on the board. After that they move alternately; Peter is first. In every move, the next player chooses two numbers, erases them and replaces both of them by their mean. Peter wins if he can achieve that the sum of suitably chosen 50 numbers is equal to the sum of the other 50 numbers. Determine whether Basil can prevent this.