2022 IMO Shortlist N5

For each $1 \leq i \leq 9$ and $T \in \mathbb{N}$, define $d_{i}(T)$ to be the total number of times the digit $i$ appears when all the multiples of 1829 between 1 and $T$ inclusive are written out in base 10. Show that there are infinitely many $T \in \mathbb{N}$ such that there are precisely two distinct values among $d_{1}(T), d_{2}(T), \ldots, d_{9}(T)$. (United Kingdom)

对于每个 $1 \leq i \leq 9$ 和 $T \in \mathbb{N}$，将 $d_{i}(T)$ 定义为当 1 到 $T$ 之间的所有 1829 的倍数以 10 为基数写出时，数字 $i$ 出现的总次数。表明存在无限多个 $T \in \mathbb{N}$，使得其中恰好有两个不同的值$d_{1}(T)、d_{2}(T)、\ldots、d_{9}(T)$。 (英国)