HMMT 2025 U28

Let $f$ be a function from nonnegative integers to nonnegative integers such that $f(0) = 0$ and

$$f(m) = f\left(\left\lfloor \frac{m}{2}\right\rfloor\right) + \left\lceil \frac{m}{2}\right\rceil^{2}$$

for all positive integers $m$ . Compute

$$\frac{f(1)}{1\cdot2} +\frac{f(2)}{2\cdot3} +\frac{f(3)}{3\cdot4} +\dots +\frac{f(31)}{31\cdot32}.$$

(Here, $\lfloor z\rfloor$ is the greatest integer less than or equal to $z$ , and $\lceil z\rceil$ is the least positive integer greater than or equal to $z$ .)