Task 6 - 200736 During one day, students from each of the grades 6, 7, and 8 came to a lending library; in total, there were 85 students. Exactly one third of the students from grade 6, exactly one third of the students from grade 7, and exactly one quarter of the students from grade 8, which amounted to a total of 26 students, borrowed books from the library series "Mathematical Student Library".

In addition, it emerged from conversations that exactly one tenth of the students from grade 7 had participated in the county mathematics olympiad.

Examine whether the numbers of students from grade 6, grade 7, and grade 8 can be uniquely determined from these data! If this is the case, determine these three numbers!
If $a, b, c$ are the numbers of students in grades 6, 7, 8 in this order, then from the given information:

$a$ and $b$ are divisible by 3, $c$ is divisible by 4, and $b$ is divisible by 10. Since 3 and 10 are coprime, $b$ is divisible by 30. Therefore, there exist natural numbers $p, q, r$ such that $a=3 p, b=30 q, c=4 r(1)$; where $p, q, r$ are also non-zero like $a, b, c$.

From (1) and the total number of students, it follows that $3 p+30 q+4 r=85$ (2); from (1) and the number of students who borrowed books from the "Mathematical Student Library," it follows that $p+10 q+r=26$ (3).

From (2), only $q=1$ or $q=2$ is possible. If $q=2$, then from (3): $p+r=6$ and from (2) further $3 p+4 r=3 p+3 r+r=3(p+r)+r=25$. Since $p+r=6$, it follows that $1 \leq p \leq 5$ and $1 \leq r \leq 5$. From $3(p+r)+r=3 \cdot 6+r=25$ it would follow that $r=7$, which contradicts $r \leq 5$.

Therefore, $q=1$ and thus $p+r=16$ and $3 p+4 r=3(p+r)+r=55$. From this, it follows that $3 \cdot 16+r=55$ and finally $r=7$ and $p=9$. This shows that from the given information, it is uniquely determined:

The numbers of students in grades 6, 7, and 8 are 27, 30, and 28, respectively.

Solutions of the third round 1980 taken from [5]

### 4.23 21st Olympiad 1981

### 4.23.1 First Round 1981, Grade 7