Belarus2022

Let us first prove that interesting numbers exist. Consider the cycle $$4\to 16\to 37\to 58\to 89\to 145\to 42\to 20\to 4$$ Each number in this cycle is obtained from the previous one by the operation described in the condition. Therefore, starting with any of the numbers in this cycle, Petya will never obtain $1$. Hence, all numbers in this cycle are interesting.

Note that for any interesting number $x$, the number $P(x)$, consisting of $x$ units in decimal notation, is also interesting, since the very first number that Petya writes on the blackboard will be equal to $x$. Moreover, $P(x)>x$ for any $x>1$. Therefore, for any interesting number $x$, there exists an interesting number greater than $x$, whence the statement follows.