# Task 3. Maximum 20 points

At the "Economics and Law" congress, a "Tournament of the Best" was held, in which more than 220 but fewer than 254 delegates—economists and lawyers—participated. Within one match, participants had to ask each other questions within a limited time and record the correct answers. Each participant played against each other exactly once. The winner of the match received one point, the loser received no points; in the event of a draw, both participants received half a point. At the end of the tournament, it turned out that in matches against economists, each participant scored half of all their points. How many lawyers participated in the tournament? Provide the smallest of all possible answers.
Solution: Let there be $\mathrm{n}$ delegates participating in the tournament, of which $\mathrm{m}$ are economists and $\mathrm{n}-\mathrm{m}$ are lawyers. All participants conducted $\mathrm{n}(\mathrm{n}-1) / 2$ matches and scored $\mathrm{n}(\mathrm{n}-1) / 2$ points. Among them, the economists competed in $\mathrm{m}(\mathrm{m}-1) / 2$ matches and scored the same number of points. The lawyers conducted $(n-m)(n-m-1)/2$ matches among themselves and scored $(n\mathrm{m})(\mathrm{n}-\mathrm{m}-1) / 2$ points while playing against each other.

Since each participant scored half of their points against economists, the economists scored half of their points against each other, meaning they scored twice as many points in total: $\mathrm{m}(\mathrm{m}-1)$.

Since the lawyers scored half of their points against economists, they also scored half of their points against each other, meaning they scored $(n\mathrm{m})(\mathrm{n}-\mathrm{m}-1)$ points in total. The total number of points scored by all participants is:

$m(m-1)+(n-m)(n-m-1)=n(n-1) / 2$.

After a series of transformations, we get:

$n^{2}-4 n m+4 m^{2}=n$

From which $(\mathrm{n}-2 \mathrm{~m})^{2}=\mathrm{n}$, meaning $\mathrm{n}$ is a perfect square of a natural number.

Given $220<\mathrm{n}<254$, we find that $\mathrm{n}=225$, then $(225-2 \mathrm{~m})^{2}=225$, from which $\mathrm{m}=$ 105 or $\mathrm{m}=120$.

The number of participants in the tournament is 225; the number of lawyers is 105, or 120. The smallest value is 105.

## Criteria.

- Correct solution and the right answer - 20 points (maximum),
- Correct solution but the wrong answer or no answer -

8 points (rounded to the nearest whole number, half the maximum),

- Error in the solution but the right answer - 7 points (rounded to the nearest whole number, half the maximum minus one),
- Incorrect or no solution, but the right answer - 2 points (under any circumstances),
- Incorrect or no solution, and the wrong answer or no answer - 0 points.