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 $a=b=c=1/3$.

Example 4: uvw Principle (very important! Must know for Tejs' Theorem in Level 4!)

Let $a,b,c>0$ and $a+b+c=3$. Your goal is show that the following expression is true:

First, by the uvw principle, we can set $b=c=x$ and $a=3-2x$. Therefore, the inequality thus reduces to

and the conclusion from here is immediate.

Example 5: Engel and Cauchy

Your job is to show that

We can simply expand the numerators and apply Engel form to the quadratic terms. Then, we can apply cauchy or nesbitt's inequalty to finish the problem!

Example 6: Ravi Substitution (very important! Must know!)

Let $x,y,z$ be the side lengths of a triangle. Your goal is to show that

We can use Ravi's substitution here:

Thus, the inequality becomes:

and the conclusion simply follows from the Engel form.

Example 7: Hölder and Power Mean (EXTREMELY IMPORTANT!)

Your job is to show that:

We can first apply Hölder's inequality:

Now, since $(1+1+1)^3=27$, the result follows.

Example 8: Jensen's Inequality (ALSO EXTREMELY IMPORTANT!)

Let $a,b,c>0$ with $a+b+c=1$. Your goal is to show that:

First, notice that since $\ln(1+x)$ is concave, Jensen gives

And multiplying this by $3$ gives the result

Example 9: Muirhead (ALSO ALSO EXTREMELY IMPORTANT!)

Your goal is to show that:

First, notice that since $(4,0,0)\succ(3,1,0)$, Muirhead gives

Then, we can combine it with

and the result follows nicely :)

Strategy Checklist

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

Practice Problems