Problem 23

Algebra Level 5

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

$$x,y,z$$ 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 $$S+H+K+A$$.

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