## APMO 2018 Problem 2

Discussion on Asian Pacific Mathematical Olympiad (APMO)
samiul_samin
Posts: 999
Joined: Sat Dec 09, 2017 1:32 pm

### APMO 2018 Problem 2

Let $f(x)$ and $g(x)$ be given by

$f(x) = \frac{1}{x} + \frac{1}{x-2} + \frac{1}{x-4} + \cdots + \frac{1}{x-2018}$

$g(x) = \frac{1}{x-1} + \frac{1}{x-3} + \frac{1}{x-5} + \cdots + \frac{1}{x-2017}$.

Prove that $|f(x)-g(x)| >2$ for any non-integer real number $x$ satisfying $0 < x < 2018$.