2000 Pan African MO Problems/Problem 1
Solve for :
Let and . Substitution results in By the Zero Product Property, or . We can divide into two cases.
Substituting back results in . Since both sides don't equal each other when , we can divide both sides by to get . Thus, , where is an integer.
Substituting back and applying a Pythagorean Identity results in . Multiplying both sides by and rearranging results in From the equation, or , where is an integer. However, both are extraneous solutions because both make either or undefined, so there are no solutions in the case.
Therefore the solutions to the equation are .
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