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 2 of 4: Apply weighted AM-GM
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In plain words

One copy of x and k-1 copies of 1 balance the expression.

(1+x)k≥kk(k−1)k−1x(1+x)^k\ge\dfrac{k^k}{(k-1)^{k-1}}x
Detailed analysis

For x>0x>0, weighted AM-GM applied to x,1,…,1x,1,\dots,1 gives (1+x)k≥kk(k−1)k−1x(1+x)^k\ge\dfrac{k^k}{(k-1)^{k-1}}x, with equality only at x=1/(k−1)x=1/(k-1).