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Maximum value of a sum

Source: Czech-Polish-Slovak Match 2015, Problem 3

June 19, 2015
algebrainequalitiesmaximum value

Problem Statement

Real numbers x,y,zx,y,z satisfy 1x+1y+1z+x+y+z=0\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+x+y+z=0 and none of them lies in the open interval (−1,1)(-1,1). Find the maximum value of x+y+zx+y+z.
Proposed by Jaromír Šimša
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