Determine all possible quadruplets (A, B, P, Q) of positive integers, with P ≤ Q, that satisfy this system of equations:

A + B = 21, and:

A/P^{2} + B/Q^{2} = 1

Prove that these are the only quadruplets that exist.

Well, obviously, P can only be 2, 3, 4.

If P = 1, then A/P^{2 }is >= 1

And A/P^{2} + B/Q^{2 }<=^{ }A/P^{2} + B/P^{2 }= 21/P^{2}

So, if P >= 5, then A/P^{2} + B/Q^{2} <= 21/25

*Edited on ***January 16, 2011, 1:32 pm**