MathLabs

Problem 2

Let n≥3n\ge3 be an integer, and let a2,a3,…,ana_2,a_3,\dots,a_n be positive real numbers such that a2a3⋯an=1a_2a_3\cdots a_n=1. Prove that (1+a2)2(1+a3)3⋯(1+an)n>nn(1+a_2)^2(1+a_3)^3\cdots(1+a_n)^n>n^n.
Step 1 of 4: Use the product constraint
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In plain words

The product condition lets all variable factors telescope later.

a2a3⋯an=1a_2a_3\cdots a_n=1
Detailed analysis

We are given a2a3⋯an=1a_2a_3\cdots a_n=1 with every ai>0a_i>0.