Question 115
Let $a$, $b$, $c$, $d \gt 0$. Prove the following equality:
$$
\sqrt{\left(a + \sqrt{\frac{bcd}{a}}\right)\left(b + \sqrt{\frac{acd}{b}}\right)\left(c + \sqrt{\frac{abd}{c}}\right)\left(d + \sqrt{\frac{abc}{d}}\right)} + 2\sqrt{abcd} \ge ab+bc+cd+da+ac+bd.
$$
(proposed by E. Enkhzaya)
$$
\sqrt{\left(a + \sqrt{\frac{bcd}{a}}\right)\left(b + \sqrt{\frac{acd}{b}}\right)\left(c + \sqrt{\frac{abd}{c}}\right)\left(d + \sqrt{\frac{abc}{d}}\right)} + 2\sqrt{abcd} \ge ab+bc+cd+da+ac+bd.
$$
(proposed by E. Enkhzaya)