CL-MO 2013 Problem 4

Consider a function f defined on the positive integers that meets the following conditions: $$f(1) = 1 , \, f(2n) = 2f(n) , \, nf(2n + 1) = (2n + 1)(f(n) + n)$$ for all $n \ge 1$ . a) Prove that $f(n)$ is an integer for all $n$ . b) Find all positive integers $m$ less than $2013$ that satisfy the equation $f(m) = 2m$ .