Problem 5

Let

a,b,c

be the side lengths of a triangle. Prove that

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

, and determine when equality occurs.

Step 1 of 5: Establish the two-variable square-root inequality

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x,y>0\Longrightarrow\sqrt{\frac{x+y}{2}}\ge\frac{\sqrt x+\sqrt y}{2}

Detailed analysis

Both sides are nonnegative. Squaring the desired inequality is equivalent to $x+y\ge2\sqrt{xy}$ , which is the arithmetic–geometric mean inequality. Equality holds exactly when $x=y$ .