Problem Ineks

Problem:
Jika $a,b,c$ bilangan real positif, buktikan bahwa :
$$\frac{a}{b^2 + c^2 + d^2} +\frac{ b}{a^2 + c^2 + d^2} + \frac{c}{a^2 + b^2 + d^2} + \frac{d}{a^2 + b^2 + c^2} \ge \frac{4}{a+b+c+d}$$



Solusi:
Dengan mengaplikasikan CS Inequality, didapat:
$$(a+b+c+d)\left(\sum_{cyc}\frac{a}{b^2+c^2+d^2}\right)\ge\left(\sum_{cyc}\sqrt{\frac{a^2}{b^2+c^2+d^2}}\right)^2$$
akan tetapi berdasarkan AM-GM, didapat:
$$\sqrt{\frac{b^2+c^2+d^2}{a^2}} \le \frac{1}{2}\left(\frac{b^2+c^2+d^2}{a^2}+1\right)=\frac{1}{2}\left(\frac{a^2+b^2+c^2+d^2}{a^2}\right)$$
sehingga:
$$
\left(\sum_{cyc}\sqrt{\frac{a^2}{b^2+c^2+d^2}}\right)^2\ge\left(\sum_{cyc}\frac{2a^2}{a^2+b^2+c^2+d^2}\right)^2=2^2$$
dengan demikian
$$\begin{aligned}(a+b+c+d)\left(\sum_{cyc}\frac{a}{b^2+c^2+d^2}\right)&\ge4\\
\left(\sum_{cyc}\frac{a}{b^2+c^2+d^2}\right)&\ge\frac{4}{(a+b+c+d)}\end{aligned}$$