Frankin B.R.

In a tournament with 20 athletes, 10 referees were involved. Each athlete played against each other once, and each match was officiated by exactly one referee. After each game, both participants took a photo with the referee. A year after the tournament, a stack of all these photos was found. It turned out that not everyone could be identified as either an athlete or a referee. How many such people could there be?
Let's call a person suspicious if we cannot determine whether they are an athlete or an referee. Note that each referee was only photographed with athletes, that is, no more than 20 people. On the other hand, each athlete was photographed with all other athletes and at least one referee, that is, no fewer than 20 people. Therefore, every suspicious person was photographed with exactly 20 people.

Note that if an athlete was photographed with exactly 20 people, then 19 of his games were judged by the same referee, meaning that the same person is present on all the photographs with him.

Let $X$ be some suspicious person. Since we can consider $X$ as an athlete, then, as mentioned above, there is some $Y$ present on all the photographs with him. These two were photographed together 19 times, so one of them must be a referee. Consequently, the third person on each of these photographs must be an athlete.

Thus, we can identify 19 athletes, and the remaining athlete is either $X$ or $Y$. All the others are referees. If we could determine the status of at least one of the pair $X$ and $Y$, we could determine the other as well, and there would be no suspicious people at all. Therefore, there are exactly two suspicious people - $X$ and $Y$.

## Answer

Two.

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