Suppose a function $f : R → R$ satisfies the following conditions: $(a) f(4xy) = 2y [f(x + y) + f(x -y)],$ for all $x, y ∈ R$ $(b) f(5) = 3$ .

Find the value of $f(2015)$ .

The answer is 1209.

**
This section requires Javascript.
**

You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.

It appears a linear function $f(x) = Ax + B$ will do the trick here. If this functional equation holds true for all $x, y \in \mathbb{R}$ , then at $y = 0$ we obtain $f(0) = 0$ as another boundary condition. Coupling this condition along with $f(5) = 3$ yields $\Rightarrow f(x) = \frac{3}{5}x$ and $f(2015) = \boxed{1209}.$