It's easier than you think . . . the "pigeonhole principle".
91 five-digit numbers are written on a blackboard.
Prove that one can find 3 numbers on the blackboard
such that the sums of their digits are equal.
Five-digit numbers have the range: .
Hence, their digit-sums have the range: .
. . That is, there are only 45 possible digit-sums.
With 90 numbers of the board, it is possible they have two each
. . of the 45 different digit-sums:
The 91st number must duplicate one of the digit-sums.
There will be 3 numbers with the same digit-sum.