quasi wrote: > >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.
As a number of people have surmised, the maximum possible value of A is sqrt(3).
Here's a proof ...
Let b = sqrt(3).
For x = y = z = b,
x^2 + y^2 + z^2 = b(xyz)
hence max(A) <= b.
To show that max(A) = b, it suffices to show that for all real numbers x,y,z