A Weighted Product Bound

Problem

Positive reals $a_1, \dots, a_{100}$ satisfy $a_i \ge a_{101-i}$ for $i = 1, \dots, 50$ . Define

x_k = \frac{k\,a_{k+1}}{a_1 + a_2 + \cdots + a_k} \qquad (k = 1, \dots, 99).

Prove that

x_1 x_2^2 \cdots x_{99}^{99} \le 1.

Answer

Solution

Difficulty	9/10
Topics	sequences, AM-GM, inequality

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