Question 3
A3 (USS 4), IMO 3. Consider the infinite sequences $\{x_n\}$ of positive real numbers with the following properties:
$x_0=1$ and for all $i\ge0$, $x_{i+1}\le x_i$.
(a) Prove that for every such sequence there is an $n\ge1$ such that $\frac{x_0^2}{x_1}+\frac{x_1^2}{x_2}+\cdots+\frac{x_{n-1}^2}{x_n}\ge3.999$.
(b) Find such a sequence for which $\frac{x_0^2}{x_1}+\frac{x_1^2}{x_2}+\cdots+\frac{x_{n-1}^2}{x_n}\lt 4$ for all $n$.
$x_0=1$ and for all $i\ge0$, $x_{i+1}\le x_i$.
(a) Prove that for every such sequence there is an $n\ge1$ such that $\frac{x_0^2}{x_1}+\frac{x_1^2}{x_2}+\cdots+\frac{x_{n-1}^2}{x_n}\ge3.999$.
(b) Find such a sequence for which $\frac{x_0^2}{x_1}+\frac{x_1^2}{x_2}+\cdots+\frac{x_{n-1}^2}{x_n}\lt 4$ for all $n$.