My solution assumed the product xyz is positive. If xyz is negative or 0, then the result is immediate because T is nonnegative.
> A = sqrt(3). > Assume x,y,z nonnegative. > > Let T = x^2+y^2+z^2, > S = x+y+z, > P = xyz. > > Express T^2-3SP as the sum of squares: > T^2-3SP = (x^2-yz)^2 + (y^2-zx)^2 + (z^2-xy)^2 > + (1/2)[(xz-yz)^2 + (xy-yz)^2 + (xy-xz)^2] > > Then T^2-3SP >= 0, > T^2 >= 3SP >= 3P^2 > T >= sqrt(3)P > An extremal example is x=y=z=sqrt(3). > > Don Coppersmith > > > Here's a nice challenge problem which I adapted > from > > a past > > competition problem ... > > > > Problem: > > > > Find, with proof, the largest real number A such > that > > > > x + y + z >= xyz > > > > implies > > > > x^2 + y^2 + z^2 >= Axyz > > > > My solution is elementary (avoids Calculus), but if > a > > method > > based on Calculus yields an easy resolution, that > > would be worth > > seeing as well. > > > > quasi