9,10
In the tournament, each chess player scored half of all their points in matches against the participants who took the last three places.
How many people in total participated in the tournament

#
For brevity, let's call the players who took the last three places "bad," and all others "good." The bad players played three games among themselves, and in these games, a total of three points were scored. According to the condition, this is half of all the points scored by the bad players; therefore, in games with the good players, the bad players scored another 3 points. However, a total of \(3(n-3)\) games were played between the bad and good players, and the same number of points were distributed (\(n\) is the total number of players). Of these, 3 points were taken by the bad players, and the rest by the good players. Therefore, in games with the bad players, the good players won \(3(n-3) - 3 = 3(n-4)\) points, and thus, the good players scored (in total) \(3(n-4)\) points in games with each other. Between the good players, there were \(\frac{1}{2}(n-3)(n-4)\) games and the same number of points were distributed. Therefore, \((n-3)(n-4) = 6(n-4)\), from which \(n=4\) or \(n=9\). The first option should be excluded, as in this case, the only good player would have scored 0 points, meaning he would not have been the first. The only solution left is \(n=9\). For nine players, this could have happened: an example of the corresponding tournament table is given below.

## Answer

9 people.