Determine all possible positive real quadruplets (p, q, r, s) satisfying the following system of equations:

pqrs = 27 + pq+ pr + ps + qr + qs + rs

p+q+r+s = 12

p+q+r+s=12 => (p+q+r+s)/4 = 3 Using AM, GM inequality we get
(pqrs)¼ ≤3 => pqrs ≤ 81 --(1)

pq+pr+ps+qr+qs+rs ≥ 6*√(pqrs)
2*(pqrs-27)≥6*√(pqrs)
substituting  √(pqrs) =x in the above equation, we get
x²-6x-27≥0
(x-9)*(x+3)≥0

 √(pqrs)≥9 => pqrs ≥81 from eq(1) pqrs=81
AM, GM conditions equality is satisfied when all of them are equal
so, p=q=r=s which implies p=q=r=s=3

Posted by Praneeth on 2007-07-26 11:23:25