MK-MO 1998 Problem 3

A triangle $ABC$ is given. For every positive numbers $p,q,r$ , let $A',B',C'$ be the points such that $\overrightarrow{BA'} = p\overrightarrow{AB}, \overrightarrow{CB'}=q\overrightarrow{BC}$ , and $\overrightarrow{AC'}=r\overrightarrow{CA}$ . Define $f(p,q,r)$ as the ratio of the area of $\vartriangle A'B'C'$ to that of $\vartriangle ABC$ . Prove that for all positive numbers $x,y,z$ and every positive integer $n$ , $\sum_{k=0}^{n-1}f(x+k,y+k,z+k) = n^3f\left(\frac{x}{n},\frac{y}{n},\frac{z}{n}\right)$ .