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/P² + B/Q² = 1

Prove that these are the only quadruplets that exist.

Simply recasting the equation as:

(a*(q-p)(q+p))/(q^2-21)=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 2011-01-17 02:08:11