JBMO-SL 2019 A5

Let $a, b, c, d$ be positive real numbers such that $a b c d=1$. Prove the inequality

$$\frac{1}{a^{3}+b+c+d}+\frac{1}{a+b^{3}+c+d}+\frac{1}{a+b+c^{3}+d}+\frac{1}{a+b+c+d^{3}} \leqslant \frac{a+b+c+d}{4}$$