AM–GM Inequalities

For any non-negative real numbers $a$ and $b$, the arithmetic mean is always greater than or equal to the geometric mean: $\frac{a+b}{2} \geq \sqrt{ab}$ with equality if and only if $a = b$. This can be generalized to more numbers, but the two-variable case is most common in olympiad problems.

Important properties

• The AM–GM inequality only applies to non-negative numbers.

• Equality holds if and only if all the numbers are equal.

• It can be used to estimate or bound expressions involving sums and products.

• The inequality can be extended to more than two numbers: $\frac{a_1 + a_2 + \cdots + a_n}{n} \geq \sqrt[n]{a_1 a_2 \cdots a_n}$ for $a_i \geq 0$.