MathDB

4

Part of 2008 Philippine MO

Problems(1)

Function Summation Identity

Source: 2008 Philippine Mathematical Olympiad Problem 4

5/6/2014
Let f:R→Rf:\mathbb{R}\rightarrow \mathbb{R} be a function defined by f(x)=20082x2008+20082xf(x)=\frac{2008^{2x}}{2008+2008^{2x}}. Prove that f(12007)+f(22007)+⋯+f(20052007)+f(20062007)=1003.\begin{aligned} f\left(\frac{1}{2007}\right)+f\left(\frac{2}{2007}\right)+\cdots+f\left(\frac{2005}{2007}\right)+f\left(\frac{2006}{2007}\right)=1003. \end{aligned}
function