Question

91 five digit numbers are written on a blackboard. How do you prove that you can find 3 numbers on the blackboard such that the sums of their digits are equal?

Answer 1

The amallest possible number sum of digits is 1 (for 10000) and the largest possible number is 45 (for (99999). If you pick 91 different numbers, the fewest matches you could have in the value of the digit sums occurs when you get every number but one from 1 to 45 twice, and the 91st 3 times.