Question 115
Let $Q_{\gt 1}$ be the set of rational numbers greater than 1. Let $f : Q_{\gt 1} \to \mathbb{Z}$ be a function that satisfies
$$
f(q) = \begin{cases} q - 3 & \text{if } q \text{ is an integer,} \\ \lceil q \rceil - 3 + f\left(\frac{1}{\lceil q \rceil - q}\right) & \text{otherwise.} \end{cases}
$$
Show that for any $a, b \in Q_{\gt 1}$ with $\frac{1}{a} + \frac{1}{b} = 1$, there always holds $f(a) + f(b) = -2$.
$$
f(q) = \begin{cases} q - 3 & \text{if } q \text{ is an integer,} \\ \lceil q \rceil - 3 + f\left(\frac{1}{\lceil q \rceil - q}\right) & \text{otherwise.} \end{cases}
$$
Show that for any $a, b \in Q_{\gt 1}$ with $\frac{1}{a} + \frac{1}{b} = 1$, there always holds $f(a) + f(b) = -2$.