3. (7 points) Participating in a chess tournament, Vasya played 52 games. According to the old scoring system (1 point for a win, $\frac{1}{2}$ point for a draw, and 0 points for a loss), he scored 35 points. How many points did he score according to the new scoring system (1 point for a win, 0 points for a draw, and -1 point for a loss)?
Answer: 18 points.

## Solution.

First method. Let Vasya win $a$ times, draw $b$ times, and lose $c$ times in the tournament. Then $a+b+c=52, a+\frac{b}{2}=35$. We need to find the value of $a-c$. From the second relation, it follows that $b=70-2a$. Then $a+(70-2a)+c=52$, from which $70+c-a=52, a-c=18$.

Second method. Under the scoring system $(1 ; \frac{1}{2} ; 0)$, Vasya scored 35 points, so under the system $(2 ; 1 ; 0)$, he would score twice as much, that is, 70 points.

Under the system $(1 ; 0 ;-1)$, Vasya loses one point in each game (compared to the system $(2 ; 1 ; 0)$). Therefore, he will score $70-52=18$ points.

## Grading Criteria.

- Any complete correct solution - 7 points.
- Correct reasoning, in which the correct answer is not obtained due to an arithmetic error, - 5 points.
- A specific case is considered, that is, specific values of the number of wins, draws, and losses that satisfy the problem's condition, and the correct answer is obtained - 2 points.
- Only the answer is provided - 0 points.