91 five-digit numbers are written on a blackboard. Prove that one can find three numbers on the blackboard such that the sums of their digits are equal.
Calculate the number of possible sums of five digits. The lowest is 1 and the highest is 45. So no matter how you pick the numbers, the sum will have to be duplicated at least twice when picking 90 numbers. The 91st will have to be a three-pete.
