Question 17
(YUG 1) Prove the inequality
$$\frac{a_1+a_3}{a_1+a_2}+\frac{a_2+a_4}{a_2+a_3}+\frac{a_3+a_1}{a_3+a_4}+\frac{a_4+a_2}{a_4+a_1}\ge4,$$
where $a_i\gt 0$, $i=1,2,3,4$.
$$\frac{a_1+a_3}{a_1+a_2}+\frac{a_2+a_4}{a_2+a_3}+\frac{a_3+a_1}{a_3+a_4}+\frac{a_4+a_2}{a_4+a_1}\ge4,$$
where $a_i\gt 0$, $i=1,2,3,4$.