USAMO 2017 Problem 6

Find the minimum possible value of $$\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4}$$ given that $a$ , $b$ , $c$ , $d$ are nonnegative real numbers such that $a+b+c+d=4$ .

Proposed by Titu Andreescu