Problem

Source: 2023 Junior Macedonian Mathematical Olympiad P3

Tags: inequalities, algebra



Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove the inequality $$ \left ( \frac{1+a}{b}+2 \right ) \left ( \frac{1+b}{c}+2 \right ) \left ( \frac{1+c}{a}+2 \right )\geq 216.$$When does equality hold? Proposed by Anastasija Trajanova