5. In a chess tournament, where each participant played against every other, three chess players fell ill and dropped out of the tournament before it reached its halfway point. A total of 130 games were played in the tournament. How many chess players participated in the tournament?
Answer: 19 people

Solution.

If 16 chess players participated in the tournament, the number of games they played should not exceed (16 x 15):2 = 120 games. Therefore, more than 16 people played in the tournament. Let's consider the following cases.

A) The tournament started with 17 participants. Then 14 of them, who finished the tournament, played (14 x 13):2 = 91 matches among themselves, and the eliminated players played 39 matches together. Therefore, someone among them played no less than 13 matches, i.e., was eliminated in the second half of the tournament. This contradicts the condition.

B) The tournament started with 18 participants. Then 15 of them, who finished the tournament, played (15 x 14):2 = 105 matches among themselves, and the eliminated players played 25 matches together. Since half of the tournament consists of 8 rounds, the eliminated participants could not have played more than 24 matches together.

C) The tournament started with 19 participants. Then 16 of them, who finished the tournament, played (16 x 15):2 = 120 matches among themselves, and the eliminated players played 10 matches together. Each of them could have been eliminated in the first half of the tournament.

D) The tournament started with 20 participants. Then 17 of them, who finished the tournament, played (17 x 16):2 = 136 matches among themselves, which is more than all the participants in the tournament.

Therefore, 19 people participated in the tournament.