4. In the Olympionov family, it is a tradition to especially celebrate the day when a person turns as many years old as the sum of the digits of their birth year. Kolya Olympionov had such a celebration in 2013, and Tolya Olympionov had one in 2014. Who is older and by how many years?
Solution. Let's determine in which year a person could be born if adding the sum of the digits of their birth year results in 2013 or 2014. It is clear that the year of such a birth cannot be later than 2014. Since the sum of the digits of each number from 1 to 2014 does not exceed \(1+9+9+9=28\), the birth year cannot be earlier than 2013-28=1985. By checking all years from 1985 to 2013, we find that Kolya could have been born in 1992 or 2010, and Tolya in 1988 or 2006.

Therefore, the possible scenarios are: Tolya is 4 years older; Tolya is 22 years older; Kolya is 14 years older.

## Criteria.

Complete solution. The solution should include a check or at least indicate the range of the check. Justifications such as "It is sufficient to check only from 1985 to 2014 because the sum of the digits is no more than 28" are not required - it is enough for the range to include the years from 1988 to 2010.

Errors. If nothing is written about the check, and it is immediately stated, "Kolya could have been born in 1992 or 2010, and Tolya in 1988 or 2006, therefore... " and the correct answer follows, this is considered a solution with an error and is scored 5 points.

Progress. If one of the possible birth years for Kolya and one for Tolya is indicated - 1 point is given. If one variant is found for one brother and both for the other, 2 points are given.

If there are "extra" birth years given, the solution is scored no more than 2 points. A correct (complete) answer without indicating the years they could have been born - 2 points.

Note. Strictly speaking, we found the difference in birth years, but the difference in ages can lie in the range from 3 to 5 years or from 21 to 23 years or from -15 to -13 years. Mentioning this is not required but is not penalized.