International Mathematical Olympiad 1982 Problem 3

InequalitiesOptimizationSeq

Consider the infinite sequences {xn}\{x_n\} of positive real numbers with the following properties:

x0=1, and for all i≥0,xi+1≤xi.x_0 = 1, \text{ and for all } i \geq 0, x_{i+1} \leq x_i.

(a) Prove that for every such sequence, there is an n≥1n \geq 1 such that

x02x1+x12x2+⋯+xn−12xn≥3.999.\frac{x_0^2}{x_1} + \frac{x_1^2}{x_2} + \cdots + \frac{x_{n-1}^2}{x_n} \geq 3.999.

(b) Find such a sequence for which

x02x1+x12x2+⋯+xn−12xn<4.\frac{x_0^2}{x_1} + \frac{x_1^2}{x_2} + \cdots + \frac{x_{n-1}^2}{x_n} < 4.