I spent some time thinking about the infamous IMO 1988/Problem 6 today:
For positive integers , if is an integer, prove that it is a square number.
After some initial false starts, I came up with this:
. We need to eliminate the , in order to have some hope of a quotient that is an integer. One way to do that is to have . This implies that . We have thus proven that the quotient, which is , is a square number.
Of course, this is not the complete solution. We might have , in which case the quotient could also be an integer. However, it turns out that the solution that I have written above is the only possible solution. Now we only need to justify that this is the only solution.