Japan Mathematical Olympiad

$22$

For a positive integer $k$, let us define the changed number of $k$ as the number obtained by switching each digit $2$ in $k$ to $3$, and each digit $3$ in $k$ to $2$. By this definition, if $l$ is the changed number of $k$, then the changed number of $l$ is $k$ itself. For a good number $m\le 1999$ and a bad number $n\le 1999$, we call $(m,n)$ a nice pair if $n$ is the changed number of $m$.

Let $m$ be a good number. Let $a$ (respectively, $b$) be the number of times the digit $2$ (respectively, $3$) appears in $m$. Then, we have $a>b$. Let $n$ denote the changed number of $m$. Then, the number of times the digit $2$ appears in $n$ is $b$, and the number of times the digit $3$ appears in $n$ is $a$. Hence, $n$ is a bad number. Furthermore, if $m\le 1999$, then we have $n\le 1999$. Note that a number cannot be both a good number and a bad number. These observations show that each good number less than or equal to $1999$ is contained in exactly one nice pair. Similarly, each bad number less than or equal to $1999$ is contained in exactly one nice pair. Therefore, the number of good numbers less than or equal to $1999$ and the number of bad numbers less than or equal to $1999$ are both equal to the number of nice pairs.

In addition, among the integers between $2000$ and $2023$, there are $22$ good numbers (excluding $2003$ and $2013$) and no bad numbers. Thus, the answer is $22$.