AoPS - Problem

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Problem

Source: Iranian National Olympiad (3rd Round) 2002

Tags: function, limit, algebra proposed, algebra

Omid Hatami

01.10.2006 08:41

$f: \mathbb R\longrightarrow\mathbb R^{+}$ is a non-decreasing function. Prove that there is a point $a\in\mathbb R$ that $f(a+\frac1{f(a)})<2f(a)$

Rust

01.10.2006 09:23

Let $f(x+\frac{1}{f(x)})>2f(x)$ for all x. Let $x_{0}=0, x_{n+1}=x_{n}+\frac{1}{f(x_{n})}$ . Then $f(x_{n})>2^{n}a, a=f(x_{0})>0$ , therefore $x_{n}<x_{0}+\sum_{k=0}^{n-1}\frac{1}{a2^{k}}<\frac{2}{a}$ . It give, that exist $x=\lim_{n}x_{n}$ and $\forall C>0 \ f(x)>C$ (contradition).