RO-TST 1996 Problem 7

Let $a\in \mathbb{R}$ and $f_1(x),f_2(x),\ldots,f_n(x): \mathbb{R} \rightarrow \mathbb{R}$ are the additive functions such that for every $x\in \mathbb{R}$ we have $f_1(x)f_2(x) \cdots f_n(x) =ax^n$ . Show that there exists $b\in \mathbb {R}$ and $i\in {\{1,2,\ldots,n}\}$ such that for every $x\in \mathbb{R}$ we have $f_i(x)=bx$ .