Problem

Source: 44th International Tournament of Towns, Junior A-Level P2, Fall 2022

Tags: number theory, Tournament of Towns



The numbers $1, 19, 199, 1999,\ldots$ are written on several cards, one card for each number. Is it possible to choose at least three cards so that the sum of the numbers on the chosen cards equals a number in which all digits, except for a single digit, are twos? Suppose you have chosen several cards so that the sum of the numbers on the chosen cards equals a number, all of whose digits are twos, except for a single digit. What can this single different digit be?