Log-Convexity in the Middle

★ 8/10Monotonicity sequences Telescoping inequality

Problem

Real numbers $x_1, x_2, \dots, x_{2018}$ satisfy

x_{n+1}^2 \le x_n x_{n+2} \quad (n = 1, \dots, 2016) \qquad\text{and}\qquad \prod_{n=1}^{2018} x_n = 1.

Prove that $x_{1009}\,x_{1010} \le 1$ .

Answer

Solution

Difficulty	8/10
Topics	Monotonicity, sequences, Telescoping, inequality

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