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If $1+ab$ divides $a^2+b^2$ , prove that the quotient is a perfect square.

Tried to use the fact that $\displaystyle\frac{a^2+b^2}{1+ab}$ is an integer but I am stuck.

MathGod
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    This is IMO 1988/6 and is asked many times before on MSE. – Bart Michels Jan 20 '14 at 16:17
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    For example here http://math.stackexchange.com/questions/94069/fraca2b21ab-is-a-perfect-square-whenever-it-is-an-integer?rq=1 –  Jan 20 '14 at 16:19
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    And here http://math.stackexchange.com/questions/28438/alternative-proof-that-a2b2-ab1-is-a-square-when-its-an-integer –  Jan 20 '14 at 16:20

2 Answers2

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It is not so elementary. It is a famous IMO Problem (6th problem of 1988 IMO).

For a solution see here.

Yiorgos S. Smyrlis
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Hint $\ $ Apply what's called Vieta jumping, but which is really descent on conics using reflections. See also this question.

Community
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Bill Dubuque
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