Question 23
Let $k$ be a positive integer and let the function $f$ be defined as follows: $$f(x)=\frac{\pi^x}{\pi^x+\pi^{2k-x}}.$$ Suppose the function $g(k)$ is defined as follows: $$g(k)=f(0)+f\left(\frac{k}{2019}\right)+f\left(\frac{2k}{2019}\right)+f\left(\frac{3k}{2019}\right)+\cdots+f\left(\frac{4037k}{2019}\right)+f(2k).$$ Find the greatest positive integer $n$ such that $g(k)\ge n$ for all $k\ge1$.