jmc

Question

Let w, x, y, and z be nonnegative numbers whose sum is 100. Find the largest possible value of wx + xy + yz.

Accepted Answer

We have that wx + xy + yz wx + xy + yz + zw = (w + y)(x + z).By AM-GM, (w + y)(x + z) ( (w + y) + (x + z)2 )^2 = 2500.Equality occurs when w = x = 50 and y = z = 0, so the largest possible value is 2500.