Problem 2, 1998

Let \(a\), \(b\), \(c\) be three distinct nonzero real numbers, and for real \(x, y \neq a\) set

\[ V(x,y)=\frac{1}{(a-x)^{2}(a-y)^{2}}\Bigl((a-b)^{2}(c-x)(c-y)-(c-a)^{2}(b-x)(b-y)\Bigr). \]

Prove that there exist expressions \(f(x)\) and \(g(y)\), the first free of \(y\) and the second free of \(x\), such that

\[ V(x,y)=\frac{1}{x-y}\bigl(f(x)-g(y)\bigr) \]

holds for all real \(x, y\) with \(x \neq y\) and \(x, y \neq a\).