CN-LEAGUE 2007 Problem 3

For positive integers $k,m$ , where $1\le k\le 5$ , define the function $f(m,k)$ as $$f(m,k)=\sum_{i=1}^{5}\left[m\sqrt{\frac{k+1}{i+1}}\right]$$ where $[x]$ denotes the greatest integer not exceeding $x$ . Prove that for any positive integer $n$ , there exist positive integers $k,m$ , where $1\le k\le 5$ , such that $f(m,k)=n$ .