Multiply In Pairs, Get Product (Posted on 2007-07-26)

	Submitted by K Sengupta
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In terms of AM- GM inequality, we obtain: (pq+ pr + ps + qr + qs + rs)/6 ≥ ((pqrs)3)1/6 = V(pqrs) or, (27+ pq+ pr + ps + qr + qs + rs)≥ 27 + 6*V(pqrs) Thus: pqrs ≥ 27 + 6*V(pqrs) Or, (Vpqrs + 3)*(Vpqrs - 9) ≥ 0 Or, Vpqrs ≥ 9, ignoring the other inequality since each of p, q, r and s are positive.......(i) Again, 12 = p+q+r+s ≥ 4*(pqrs)1/4 Or, V(pqrs)≤ 3^2 = 9........(ii) From (i) and (ii), we obtain V(pqrs) = 9, so that from AM- GM inequality, it follows that each of the four numbers must be equal. This is possible iff p=q=r=s=3, which is the only possible solution to the given problem.

	Subject	Author	Date
	Solution	Praneeth	2007-07-26 11:23:25