Problem 23

$(x+y+z) \left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \right) \ge s \left( \frac{x}{y+z} + \frac{y}{z+x} + \frac{z}{x+y} \right)$

$x,y,z$ are real numbers in the interval $[1, 2]$ . Let $S$ be the maximum value of $s$ such that the inequality holds for all such $x,y,z$ , and suppose that in this case, equality is achieved when $Hx = Ky = Az$ . Compute $S+H+K+A$ .

This problem was in Vietnam TST a few year back.

This problem is in this Set .