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.
Simply recasting the equation as:
(a*(qp)(q+p))/(q^221)=p^2
gives solutions {a,p,q} at {5,3,6}{12,4,6}{37,6,24}{41,6,12}{48,6,9}{85,9,36}{101,9,18}...etc. of which only the first two have a less than 21, compliant with the stipulations of the problem.

Posted by broll
on 20110117 02:08:11 