Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 6.2 National olympiad Prove it

Let $3.21 a, b, c, d$ be non-negative real numbers. Prove
$\begin{array}{c} \sqrt{1+\frac{7 a}{b+c+d}}+\sqrt{1+\frac{7 b}{c+d+a}}+\sqrt{1+\frac{7 c}{d+a+b}}+ \\ \sqrt{1+\frac{7 d}{a+b+c}} \geqslant 4 \sqrt{\frac{10}{3}} \end{array}$

Solution

By the homogeneity of the inequality, without loss of generality, assume $a+b+c+d=4$ . Let $f(x)=\sqrt{1+\frac{7 x}{4-x}}$ , then we have
$f(x) \geqslant \frac{1}{3} \sqrt{\frac{2}{15}}(8+7 x)$

The above inequality holds because
$\frac{14(x-1)^{2}(2+7 x)}{135(4-x)} \geqslant 0$

Substituting $a, b, c, d$ into $f(x)$ respectively and adding them up yields the desired inequality.

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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.