IIT JEE , AIEEE Forum - Mathematics , Algebra

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The denominator is

, so

. If n = 1, then m must be even, in other words, we have the solution (m, n) = (2k, 1).

So assume n > 1. Put

. Then we have a quadratic equation for m, namely

. This has solutions

, where N is the positive square root of

. Since

, N is certainly real. But the sum and product of the roots are both positive, so both roots must be positive. The sum is an integer, so if one root is a positive integer, then so is the other.

The larger root

is greater than

, so the

. But note that if

, then since h > 0, we must have the denominator

smaller than the numerator and hence

. So for the smaller root we cannot have

. But

must be non-negative (since h is positive), so

for the smaller root. Hence

. Now

, so

. Thus n must be even. Put

and we get the solutions

and

.

Hence Solutions are :

ps. IMO-2003 Dont rate me. This is the official solution.

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