Question 107
Let $\mathbb{Z}^+$ denote the set of all positive integers. Find all surjective functions $f: \mathbb{Z}^+ \times \mathbb{Z}^+ \to \mathbb{Z}^+$ that satisfy all of the following conditions: for all $a, b, c \in \mathbb{Z}^+$,
(i) $f(a, b) \le a + b$
(ii) $f(a, f(b, c)) = f(f(a, b), c)$
(iii) Both $\binom{f(a, b)}{a}$ and $\binom{f(a, b)}{b}$ are odd numbers (where $\binom{n}{k}$ denotes the binomial coefficient).
(i) $f(a, b) \le a + b$
(ii) $f(a, f(b, c)) = f(f(a, b), c)$
(iii) Both $\binom{f(a, b)}{a}$ and $\binom{f(a, b)}{b}$ are odd numbers (where $\binom{n}{k}$ denotes the binomial coefficient).