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Question

Let x, y, and z be positive real numbers such that xyz(x + y + z) = 1. Find the minimum value of (x + y)(y + z).

Accepted Answer

We can write (x + y)(y + z) as xz + y(x + y + z). By AM-GM, xz + y(x + y + z) 2 (xz)y(x + y + z) = 2 xyz(x + y + z) = 2.Equality holds when xz = y(x + y + z) = 1 and xyz(x + y + z) = 1. For example, we can take x = 1, y = 2 - 1, and z = 1. Hence, the minimum value is 2.