Problem

Source: 2017 USAMO #6

Tags: inequalities, AMC, USA(J)MO, USAMO(2017), Tangent line trick, Hi



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