Let $a, b, c$ and $d$ be positive numbers such that $a+b+c+d =4$, prove that
$$ \frac{4}{abcd} \ge \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} $$
I am supposed to prove my using AM-GM, so I tried
$$ a + b+ c+ d \ge 4(abcd)^{1/4} $$ $$ 1 \ge abcd$$
also by AM GM $$\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} \ge 4 $$ The only way I get the desired result is by dividing the first inequality by second and cross multiplying, but two inequalities can't be divided without sign change.