3-rd 1. In a chess tournament ${ }^{3}$, two students from grade VII and a certain number of students from grade VIII participated. The two seventh-graders scored 8 points, and each of the eighth-graders scored the same number of points. How many eighth-graders participated in the tournament? Find all solutions.
Let $x$ - the number of eighth graders, $y$ - the number of points scored by each eighth grader. By calculating the total points scored by all participants in the tournament in two ways, we arrive at the equation

$$
x y+8=\frac{(x+2)(x+1)}{2}
$$

i.e.

$$
2 y=\frac{(x+2)(x+1)-16}{x}=x+3-\frac{14}{x}
$$

Therefore, $x$ takes one of the values $1,2,7,14$. The values 1 and 2 are excluded, as in these cases the number $y$ would be negative. The problem has two answers: $x=7$ and $x=14$.