Maths Olympiad Prep

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Algebra Difficulty 5.5 AIME, harder Prove it

Prove: For the positive real numbers a, b, c, the inequality

a(a+b)(a+c)+b(b+a)(b+c)+c(c+a)(c+b)≤32. \frac{a}{\sqrt{(a+b)(a+c)}}+\frac{b}{\sqrt{(b+a)(b+c)}}+\frac{c}{\sqrt{(c+a)(c+b)}} \leq \frac{3}{2} .

holds.

Solution

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.