1987 USAMO Problems/Problem 1
Find all solutions to , where m and n are non-zero integers.
Expanding both sides, Note that can be canceled and as , can be factored out. Writing this as a quadratic equation in : . The discriminant equals , which we want to be a perfect square. Miraculously, this factors as . This is square iff (if and only if) is square. It can be checked that the only nonzero that work are . Finally, plugging this in and discarding extraneous roots gives all possible ordered pairs as .
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