Overview

At AIME level, inequalities need structure. You will often combine AM-GM with clever substitutions or homogenization.

Key Ideas

• Cauchy in Engel form: $\sum \frac{a^2}{b} \ge \frac{(a+b+c)^2}{b+c+a}$.
• Jensen applies to convex functions; start with simple cases.
• Look for symmetry and equality cases to guide algebra.

Worked Example

For $a,b>0$, show $\frac{a^2}{b}+\frac{b^2}{a}\ge a+b$.

By Cauchy, $\frac{a^2}{b}+\frac{b^2}{a}\ge \frac{(a+b)^2}{a+b}=a+b$.

Practice Problems