Problem

Source: Iran MO Third Round 2021 A1

Tags: algebra, inequalities



Positive real numbers $a, b, c$ and $d$ are given such that $a+b+c+d = 4$ prove that $$\frac{ab}{a^2-\frac{4}{3}a+\frac{4}{3}} + \frac{bc}{b^2-\frac{4}{3}b+ \frac{4}{3}} + \frac{cd}{c^2-\frac{4}{3}c+ \frac{4}{3}} + \frac{da}{d^2-\frac{4}{3}d+ \frac{4}{3}}\leq 4.$$