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For any real number x, we have  ≥ 0. Equality holds precisely when x = 0.

This simple inequality has interesting consequences. For instance:

Arithmetic Mean – Geometric Mean inequality (AM-GM)

For any positive reals a₁, a₂, …, aₙ,

Equality holds precisely when the aᵢs are all equal.

The case of n = 2 can be seen by considering


(SMO 2013)     Let x₁ and x₂ be real numbers satisfying xx₂ = 2013. What’s the minimum value of (x₁ + x₂)²?

Solution: By AM-GM,

Equality holds when x₁ = x₂ = √2013, hence the minimum value is 8052.



Weighted Power Mean Inequality     Let a₁, a₂, …, aₙ and w₁, w₂, …, wₙ be positive reals with w₁ + w₂ + … + wₙ = 1. For any real number r, define

If r > s, then P(r) ≥ P(s), with equality precisely when all the aᵢs are equal.

Rearrangement Inequality     Let a₁ ≥ a₂ ≥ … ≥ aₙ and b₁ ≥ b₂ ≥ … ≥ bₙ be two non-increasing sequences of reals of the same length. If c₁, c₂, …, cₙ is any rearrangement of b₁, b₂, …, bₙ, then

Cauchy-Schwarz Inequality     For any real aᵢ s and bᵢ s, we have

Equality holds if and only if a₁/b₁ = a₂/b₂ = … = aₙ/b


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