AM-GM Inequality

The AM-GM inequality: For non-negative reals $a_1, \ldots, a_n$: $$\frac{a_1 + a_2 + \cdots + a_n}{n} \geq \sqrt[n]{a_1 \cdot a_2 \cdots a_n}$$ Equality holds if and only if $a_1 = a_2 = \cdots = a_n$. When to use: Optimization problems — "minimize $x + \frac{1}{x}$ for $x > 0$." Apply AM-GM: $x + \frac{1}{x} \geq 2\sqrt{x \cdot \frac{1}{x}} = 2$, with equality at $x = 1$. Power move: Rewrite the expression as a sum of terms whose product is constant (or vice versa), then apply AM-GM. This is a fundamental technique. Use it before reaching for Lagrange multipliers or calculus-based optimization. Many contest optimization problems are AM-GM in disguise.