## Subject II

In a chess tournament, at least 10 students participate, including 2 students from the 7th grade, with the rest being students from the 8th grade. Each participant plays exactly one game with each of the others. The two students from the 7th grade scored a total of 8 points, and the students from the 8th grade have equal scores. Knowing that one point is awarded for a win, 0.5 points for a draw, and 0 points for a loss, how many participants are in the 8th grade?
Subject II

In a chess tournament, at least 10 students participate, including 2 students from the 7th grade, with the rest being students from the 8th grade. Each participant plays exactly one game with each of the others. The two 7th grade students scored a total of 8 points, and the 8th grade students scored equal points. Knowing that one point is awarded for a win, 0.5 points for a draw, and 0 points for a loss, how many participants are in the 8th grade?

GRADING SCALE

Let \( n \) be the number of 8th grade students and \( m \) be the number of points each 8th grade student scored. Therefore, the total number of points scored in the tournament is \( n \cdot m + 8 \).

On the other hand, this number of points can also be calculated as follows: \( n + 2 \) people, each playing with \( n + 1 \) people, so the number of points is \( \frac{(n+2)(n+1)}{2} \) (the number of points is equal to the number of games).

Therefore, \( n \cdot m + 8 = \frac{(n+2)(n+1)}{2} \Leftrightarrow n(n+3-2m) = 14 \)

Since \( 14 = 1 \cdot 2 \cdot 7 \), the following cases are possible:

1) \( n = 1, m = -5 \) - false
2) \( n = 2, m = -1 \) - false
3) \( n = 7, m = 4 \) - false (there are more than 10 students)
4) \( n = 14, m = 8 \) - true

![](https://cdn.mathpix.com/cropped/2024_06_07_2634b5d72a53ef1bea90g-4.jpg?height=211&width=457&top_left_y=43&top_left_x=228)

MINISTRY OF EDUCATION

NATIONAL

## MATHEMATICS OLYMPIAD LOCAL STAGE  16.02.2014  GRADING SCALE AND MARKING

7th GRADE