Learn/Algebra/AM-GM Inequality

Algebra • Topic 24

AM-GM Inequality

The Arithmetic Mean–Geometric Mean Inequality (AM-GM) is one of the most fundamental inequalities in olympiad mathematics. It states that for any non-negative real numbers, their arithmetic mean is always greater than or equal to their geometric mean.

Statement

Theorem (AM-GM Inequality)

For non-negative real numbers

a_{1}, a_{2}, \dots, a_{n}

\frac{a _{1} + a _{2} + \dots + a _{n}}{n} \geq n a_{1} \cdot a_{2} \dots a_{n}

with equality if and only if $a_{1} = a_{2} = \dots = a_{n}$ .

Special Cases

Corollary (Two-Variable AM-GM)

For non-negative reals

a, b

\frac{a + b}{2} \geq ab

or equivalently: $a + b \geq 2 ab$

Corollary (Three-Variable AM-GM)

For non-negative reals

a, b, c

\frac{a + b + c}{3} \geq 3 ab c

Key Techniques

1. Direct Application

When you see a sum that you want to bound below, or a product you want to bound above, AM-GM often applies directly.

Example (Product Constraint)

Prove that for positive reals

x, y, z

with

x yz = 1

x + y + z \geq 3

Proof. By AM-GM:

\frac{x + y + z}{3} \geq 3 x yz = 31 = 1

Therefore $x + y + z \geq 3$ . Equality holds when $x = y = z = 1$ . ∎

2. Weighted AM-GM

Theorem (Weighted AM-GM)

For non-negative reals

a_{1}, \dots, a_{n}

and positive weights

w_{1}, \dots, w_{n}

with

\sum w_{i} = 1

w_{1} a_{1} + w_{2} a_{2} + \dots + w_{n} a_{n} \geq a_{1}^{w_{1}} a_{2}^{w_{2}} \dots a_{n}^{w_{n}}

3. Substitution Trick

Often we need to introduce auxiliary variables or rewrite expressions to apply AM-GM effectively.

Example (Minimization)

Minimize

x + \frac{1}{x}

for

x > 0

Proof (Solution). By AM-GM:

x + \frac{1}{x} \geq 2 x \cdot \frac{1}{x} = 2

Equality when $x = \frac{1}{x}$ , i.e., $x = 1$ . ∎

Common Mistakes

Warning (Common Pitfalls)

1. Forgetting non-negativity: AM-GM only works for non-negative numbers.

Missing equality condition: Always check when equality holds.
Unbalanced terms: The terms in AM-GM must multiply to a constant for a useful bound.

Tip (Pro Tip)

When the constraint is a product (like

x yz = k

), AM-GM on the sum gives a lower bound. When the constraint is a sum (like

x + y + z = k

), AM-GM on the product gives an upper bound.

Practice Problems

Exercise (Problem 1)

Prove that for positive reals

a, b, c

(a + b) (b + c) (c + a) \geq 8 ab c

Exercise (Problem 2)

a, b, c > 0

and

a + b + c = 3

, find the minimum value of

\frac{1}{a} + \frac{1}{b} + \frac{1}{c}

Exercise (Problem 3 (IMO 1964))

Prove that for

a, b, c

sides of a triangle:

a^{2} (b + c - a) + b^{2} (c + a - b) + c^{2} (a + b - c) \leq 3 ab c