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

Source: Switzerland - Swiss TST 2001 p2

February 18, 2020
inequalitiesGeometric Inequalities

Problem Statement

If a,ba,b, and cc are the sides of a triangle, prove the inequality a+b−c+c+a−b+b+c−a≤a+b+c\sqrt{a+b-c}+\sqrt{c+a-b}+\sqrt{b+c-a } \le \sqrt{a}+\sqrt{b}+\sqrt{c}. When does equality occur?
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