DK-GEORGMOHR 2019 Problem 4

Georg writes a positive integer $a$ on a blackboard. As long as there is a number on the blackboard, he does the following each day: $\bullet$ If the last digit in the number on the blackboard is less than or equal to $5$ , he erases that last digit. (If there is only this digit, the blackboard thus becomes empty.) $\bullet$ Otherwise he erases the entire number and writes $9$ times the number. Can Georg choose $a$ in such a way that the blackboard never becomes empty?