2023 IMO Shortlist N3

For positive integers $n$ and $k \geq 2$ define $E_{k}(n)$ as the greatest exponent $r$ such that $k^{r}$ divides $n$ !. Prove that there are infinitely many $n$ such that $E_{10}(n)>E_{9}(n)$ and infinitely many $m$ such that $E_{10}(m)<E_{9}(m)$. (Brazil) $\mathbf{N 4 .}$ Let $a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n}$ be $2 n$ positive integers such that the $n+1$ products $$\begin{gathered} a_{1} a_{2} a_{3} \cdots a_{n}, \\ b_{1} a_{2} a_{3} \cdots a_{n} \\ b_{1} b_{2} a_{3} \cdots a_{n}, \\ \vdots \\ b_{1} b_{2} b_{3} \cdots b_{n} \end{gathered}$$ form a strictly increasing arithmetic progression in that order. Determine the smallest positive integer that could be the common difference of such an arithmetic progression. (Canada)

对于正整数 $n$ 和 $k \geq 2$，将 $E_{k}(n)$ 定义为最大指数 $r$，使得 $k^{r}$ 除以 $n$!。证明有无数个 $n$ 使得 $E_{10}(n)>E_{9}(n)$ 和无限多个 $m$ 使得 $E_{10}(m)<E_{9}(m)$。 (巴西)$\mathbf{N 4 .}$设$a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n}$为$2 n$个正整数，使得$n+1$的乘积$$\begin{gathered} a_{1} a_{2} a_{3} \cdots a_{n}, \\ b_{1} a_{2} a_{3} \cdots a_{n} \\ b_{1} b_{2} a_{3} \cdots a_{n}, \\ \vdots \\ b_{1} b_{2} b_{3} \cdots b_{n} \end{gathered}$$ 按此顺序形成严格递增的算术级数。确定可能是此类算术级数的公差的最小正整数。 (加拿大)