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id: inequalities-olympiad title: Advanced Inequalities author: USAMO Guide Team — Aayush Bathija description: Techniques in homogenization, substitution, and inequality proofs. prerequisites: ["inequalities-advanced", "proof-writing-basics"]

Overview

Inequalities at the olympiad level demand structured reasoning rather than brute-force computation. A central skill is selecting transformations and substitutions that allow the application of standard inequalities in a clean and effective way.

Key Ideas

* Homogenize expressions to ensure all terms have the same degree. * Use substitutions such as $a = xy$, $b = yz$, $c = zx$ or $a = \frac{x}{y},; b = \frac{y}{z},; c = \frac{z}{x}$ in symmetric settings. * Identify equality cases early to guide the direction of the proof.

Core Skills

Homogenization

In many inequalities, given conditions can make direct approaches cumbersome. In such cases, homogenization is a key technique.

Example

At first glance, this inequality may seem difficult to approach. A useful observation is that most classical inequalities (such as AM-GM or Cauchy-Schwarz) are naturally applied to homogeneous expressions.

An expression is called homogeneous if all its terms have the same total degree. This property is powerful because it allows scaling: variables can be multiplied by a constant without changing the structure of the inequality.

To homogenize, rewrite the inequality using the condition $abc = 1$:

Now both sides have degree 2, and the inequality follows immediately from AM-GM:

Summing yields the desired result. $\square$

Substitution

While homogenization is effective, substitutions can often streamline the process.

Common substitutions include:

* If $abc = 1$, let

* For expressions involving reciprocals, consider

* In symmetric expressions, substitutions such as

can simplify the structure and reveal hidden symmetry.

Choosing the right substitution is often guided by experience and by examining the form of the inequality.

Proof Techniques

To prove an inequality of the form $A \ge C$, a common strategy is to construct a chain:

Finding appropriate intermediate expressions is often the central challenge.

Another key technique is analyzing equality cases:

* If equality occurs when $a = b = c$, this strongly suggests using symmetric inequalities such as AM-GM. * If equality depends on proportional relationships, inequalities like Cauchy-Schwarz may be more appropriate.

It is often helpful to determine equality conditions at the outset, as they provide direction for the proof.

Common Pitfalls

* Ignoring or failing to check equality cases * Reversing inequality directions during transformations * Overlooking domain constraints (e.g., positivity conditions)

Practice Problems