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\sqrt{a+b-c}+\sqrt{a-b+c}+\sqrt{-a+b+c} \leq \sqrt{a}+\sqrt{b} + \sqrt{c}

Source: Polish MO Recond Round 1983 p2

September 9, 2024
algebrainequalities

Problem Statement

There are three non-negative numbers a,b,c a, b, c such that the sum of each two is not less than the remaining one. Prove that a+b−c+a−b+c+−a+b+c≤a+b+c. \sqrt{a+b-c} + \sqrt{a-b+c} + \sqrt{-a+b+c} \leq \sqrt{a} + \sqrt{b} + \sqrt{c}.
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