APMO 2018 Problem 2
Posted: Thu Jan 10, 2019 11:04 pm
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 $0 < x < 2018$.
$f(x) = \frac{1}{x} + \frac{1}{x-2} + \frac{1}{x-4} + \cdots + \frac{1}{x-2018}$
$g(x) = \frac{1}{x-1} + \frac{1}{x-3} + \frac{1}{x-5} + \cdots + \frac{1}{x-2017}$.
Prove that $|f(x)-g(x)| >2$ for any non-integer real number $x$ satisfying $0 < x < 2018$.