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.

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