Question

Using the Pigeon-hole principle...

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.

Answer 1

I don't know what the pidgeon-hole principle is, but I imagine the solution will go something like this:

The largest five-digit number is 99999, the sum of whose digits is 45.
The smallest five-digit number is 10000, the sum of whose digits is 1.

You can find any sum between those values by some choice of 45 five-digit numbers, so if you choose 91 such numbers, then whatever they are three of them must have the same sum, because there are two lots of 45 numbers plus 1 in 91.