# Task 3. Maximum 20 points

At the conference "Economics of the Present," an intellectual tournament was held, in which more than 198 but fewer than 230 scientists, including doctors and candidates of sciences, participated. Within one match, participants had to ask each other questions and record correct answers within a certain time. 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 doctors of sciences, each participant scored half of all their points. How many candidates of sciences participated in the tournament? Provide the smallest of all possible answers.
Solution: Let there be $\mathrm{n}$ scientists participating in the tournament, of which $\mathrm{m}$ are doctors and $\mathrm{n}-\mathrm{m}$ are candidates of science. All participants conducted $\mathrm{n}(\mathrm{n}-1) / 2$ matches and scored $\mathrm{n}(\mathrm{n}-1) / 2$ points. Among them, the doctors of science competed in $\mathrm{m}(\mathrm{m}-1) / 2$ matches and scored the same number of points. The candidates conducted $(n-m)(n-m-1)/2$ matches among themselves and scored $(n\mathrm{m})(\mathrm{n}-\mathrm{m}-1) / 2$ points when playing against each other.

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

Since the candidates of science scored half of their points against the doctors of science, they also scored half of their points when competing among themselves, and in total they scored $(n-m)(n-m-1)$ points. 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 $(\mathrm{n}-2 \mathrm{~m})^{2}=\mathrm{n}$, that is, $\mathrm{n}$ is the square of a natural number.

Given $198<\mathrm{n}<230$, 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 candidates of science is 105, or 120. The smallest value is 105.

## Criteria.

- A correct solution to the problem and the correct answer are provided - 20 points (maximum),
- A correct solution to the problem, but an incorrect answer or no answer - 8 points (rounded to the nearest whole number, half the maximum),
- An error in the solution, but the correct answer is provided - 7 points (rounded to the nearest whole number, half the maximum minus one),
- The solution is incorrect or absent, but the correct answer is provided - 2 points (under any circumstances),
- The solution is incorrect or absent, and an incorrect answer or no answer is provided - 0 points.