5. The grandmother has three grandchildren. If a grandchild finished the first grade, the grandmother gave them one book; if they finished the second grade, she gave them two books; if the third grade, then three books, and so on. The books received as gifts over the years were placed on one shelf. Currently, there are 23 books on the shelf. It is known that one of the grandchildren is at least two years older than the others. Which grade did this grandchild finish?
Answer. Sixth grade.

Solution. From the condition of the problem, it follows that if the grandson has finished the second grade, then there are $1+2=3$ of his books on the shelf, if he has finished the third grade, then $1+2+3=6$ of his books, and so on. For convenience, let's make a table.

| What grade the grandson has finished | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| How many of his books are on the shelf | 1 | 3 | 6 | 10 | 15 | 21 | 28 |

It is obvious that none of the grandchildren could have finished the seventh grade, as then there would be no fewer than 28 books on the shelf. Since there are 23 books on the shelf, we need to represent the number 23 as the sum of either two numbers from the second row of the table (for the case where one of the grandchildren has not yet finished the first grade) or three such numbers. After a brief enumeration, we get that there are only two ways to represent 23: $23=3+10+10=1+1+21$. In the first case, one grandson has finished the second grade and two grandchildren have finished the fourth grade. This contradicts the fact that one of the grandchildren is at least 2 years older than the others. In the second case, two grandchildren have finished the first grade and one grandson has finished the sixth grade, which satisfies all the conditions of the problem. Thus, the oldest grandson has finished the sixth grade.

## Grading criteria.

- Any complete correct solution - 7 points.
- Generally correct solution with minor gaps in justifications 5-6 points.
- Correct answer obtained, but the case where there are 3 books of the youngest grandson and 10 books each for the other two grandchildren is not considered and the condition about the oldest grandson is not used - 4 points.
- Found how many books the grandson has if he has finished the second, third, and so on grades, but no further progress - 1 point.
- Only the answer is provided - 0 points.

Maximum score for all completed tasks - 35.