Hello, ILoveMaths07!

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 only45possible 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.

Therebe 3 numbers with the same digit-sum.will