International Mathematical Olympiad 1988 Problem 3

A function $f$ is defined on the positive integers by

$$\begin{aligned} f(1) &= 1, \quad f(3) = 3, \\ f(2n) &= f(n), \\ f(4n + 1) &= 2f(2n + 1) - f(n), \\ f(4n + 3) &= 3f(2n + 1) - 2f(n), \end{aligned}$$

for all positive integers $n$.

Determine the number of positive integers $n$, less than or equal to 1988, for which $f(n) = n$.