Answers to chess problems of grade 5 children

Threads:

There are 40 players playing chess in a round robin format. Each player plays against the other players exactly once. Each win gets 1 point, a draw gets 0.5 points and a loss gets 0 points.

What is the maximum number of participating players who can score 24 points or more?

Answer:

Note that even if the outcome of a game is “Win – Lose or Draw”, the total score of two players received in a game of chess is always 1 point.

Because there are 40 players playing in a circle, at the end of the tournament, the total score of 40 players is equal to the total number of matches and equal (40 x 39)/2 = 780 (points). From that, it follows that the number of players with 24 or more points does not exceed 780/24 = 32.5, that is, does not exceed 32 players.

We will prove that it is impossible for 32 players to score 24 or more.

Indeed, suppose there are 32 players scoring 24 or more. We have these 32 players competing against each other and will score at least 24 – 8 = 16 points (happens when 32 players win the remaining 8 players).

On the other hand, when 32 players compete against each other, the number of players with 16 or more points does not exceed ((32 x 31)/2) : 16 = 31. This leads to a contradiction and proves the assumption wrong.

From that, it is inferred that there is a maximum of 31 players with 24 points or more. This result can be shown if the 31 players are tied and each player wins all the remaining 9 players. Then 31 players together have 30 x 0.5 + 9 = 24 points.

Conclude: There are up to 31 participating players who can score 24 points or more.

Tran Phuong