2022 IMO Shortlist C9

Let $\mathbb{Z}_{\geq 0}$ be the set of non-negative integers, and let $f: \mathbb{Z}_{\geq 0} \times \mathbb{Z}_{\geq 0} \rightarrow \mathbb{Z}_{\geq 0}$ be a bijection such that whenever $f\left(x_{1}, y_{1}\right)>f\left(x_{2}, y_{2}\right)$, we have $f\left(x_{1}+1, y_{1}\right)>f\left(x_{2}+1, y_{2}\right)$ and $f\left(x_{1}, y_{1}+1\right)>f\left(x_{2}, y_{2}+1\right)$. Let $N$ be the number of pairs of integers $(x, y)$, with $0 \leq x, y<100$, such that $f(x, y)$ is odd. Find the smallest and largest possible value of $N$.

令 $\mathbb{Z}_{\geq 0}$ 为非负整数集合，并令 $f: \mathbb{Z}_{\geq 0} \times \mathbb{Z}_{\geq 0} \rightarrow \mathbb{Z}_{\geq 0}$ 为双射，每当 $f\left(x_{1}, y_{1}\right)>f\left(x_{2}, y_{2}\right)$，我们有 $f\left(x_{1}+1, y_{1}\right)>f\left(x_{2}+1, y_{2}\right)$ 和 f\left(x_{1}, y_{1}+1\right)>f\left(x_{2}, y_{2}+1\右)。令$N$ 为整数对$(x, y)$ 的数量，其中$0 \leq x, y<100$，使得$f(x, y)$ 为奇数。求 $N$ 的最小和最大可能值。