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.

Core Skills

Homogenize

If an inequality is not homogeneous, scale variables to introduce a fixed sum or product.

Use Substitutions

Try $a=xy$, $b=yz$, $c=zx$ or $a=x+y$ to simplify symmetric expressions.

Track Equality

Check when equality holds and use it to guide substitutions or case splits.

Worked Example

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

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

More Examples

Example 1: AM-GM Chain

For $a,b,c>0$, show $a+b+c \ge 3\sqrt[3]{abc}$.

Apply AM-GM to three variables.

Example 2: Engel Form

Show $\sum \frac{a^2}{b} \ge \frac{(a+b+c)^2}{a+b+c}$.

Apply Engel form directly.

Example 3: Symmetry

If $a+b+c=1$, find the minimum of $a^2+b^2+c^2$.

It is $1/3$ at $a=b=c=1/3$.

Strategy Checklist

• Homogenize if needed.
• Use symmetry to reduce variables.
• Track equality conditions.

Practice Problems