AM-GM inequality

In today's codeforces match, there is a problem ask for x,y,z such that
x+y+z = S for a given S
to maximize x^a * y^b * z^c

peter50216 points out AM-GM inequality applies to this problem.

for a, b, c > 0
S = x+y+z = a*x/a + b*y/b + c*z/c >= ((x/a)^a * (y/b)^b * (z/c)^c)^(1/abc)
and equality holds when x/a = y/b = z/c = (x+y+z)/(a+b+c)