Problem:
A row of fifty coins with integer denominations is given, such that the sum of the denominations is odd. Alice and Bob alternate taking either coin at the left end of the row or the right end of the row, with Alice playing first. Prove that Alice can always ensure she gets more than half the money.
Solution:
Color the coins alternatively black and white. Since is even, on Alice's turn, the coins at either end of the row are different colors.
Thus Alice could guarantee getting all of the black coins, she could also guarantee getting all of the white coins. Since either the sum of the black coins is more than the sum of the white coins, or vice-versa (they are not equal since the sum is odd), Alice can guarantee getting more money than Bob.