2. (12 points) In a family, there are four children of different ages. Their total age is 31 years. Four years ago, the total age of all the children in the family was 16 years, 7 years ago it was 8 years, and 11 years ago it was 1 year. How old are the children at present? (Age is always expressed as a whole number of years.)
Answer: $3, 6, 10, 12$ years.

Solution. Let's start from the end. 11 years ago, there was one child who was 1 year old. We will call this child the oldest. 7 years ago, the oldest was 5 years old, and the second child was either 3 years old or the oldest was 5 years old, the second was 2 years old, and the third was 1 year old. There are no more options due to the fact that all children are of different ages. 4 years ago, the oldest was 8 years old, the second was 6 years old, and the third was 2 years old, or the oldest was 8 years old, the second was 5 years old, and the third was 4 years old. However, according to the condition, the total age is 16 years, so we exclude the second option. Therefore, at present, the oldest is 12 years old, the second is 10 years old, the third is 6 years old, and the youngest is 3 years old.

Grading criteria. A correct answer obtained with justification - 12 points. There are arithmetic errors in the solution with the correct approach - minus 2 points. The second case in the solution is not considered - minus 3 points.