Question

В школе состоялся круговой шахматный турнир среди 25 ребят, то есть каждые двое сыграли одну партию в шахматы. После турнира каждый из игроков заявил, что он выиграл либо 5, либо 10, либо 20 партий, 3 свел в ничью, а остальные партии проиграл. Докажите, что кто-то из участников турнира ошибся в своем утверждении. СРОЧНО,ДАМ МНОГО БАЛЛОВ​

Answer 1

Ответ:
25 +5+ 20 + 3 =53

Answer 2

Problem Analysis

We are given that there was a circular chess tournament among 25 participants, with each pair playing one game of chess. After the tournament, each player claimed to have won either 5, 10, or 20 games, tied 3 games, and lost the rest. We need to prove that at least one of the participants made a false claim.

Proof by Contradiction

Let's assume that all participants made accurate claims about their wins, ties, and losses. We will show that this assumption leads to a contradiction.

Total Number of Games

Since each pair of participants played one game, the total number of games played in the tournament can be calculated using the formula for the number of combinations: C(n, 2) = n! / (2! * (n-2)!), where n is the number of participants. In this case, n = 25, so the total number of games played is C(25, 2) = 25! / (2! * 23!) = 300.

Total Number of Wins

According to the claims made by the participants, the total number of wins can be calculated by summing up the number of wins claimed by each participant. Let's denote the number of wins claimed by each participant as w1, w2, ..., w25. The total number of wins is then w1 + w2 + ... + w25.

Total Number of Losses

Similarly, the total number of losses can be calculated by subtracting the number of wins claimed by each participant from the total number of games played. Let's denote the number of losses claimed by each participant as l1, l2, ..., l25. The total number of losses is then (300 - w1) + (300 - w2) + ... + (300 - w25).

Total Number of Ties

The total number of ties is given as 3.

Contradiction

According to our assumption, the total number of wins, losses, and ties should add up to the total number of games played. Therefore, we have the equation:

(w1 + w2 + ... + w25) + (l1 + l2 + ... + l25) + 3 = 300

Simplifying the equation, we get:

2(w1 + w2 + ... + w25) + (l1 + l2 + ... + l25) = 297

Since the left side of the equation is even, the right side must also be even. However, the sum of 297 is odd, leading to a contradiction.

Conclusion

Our assumption that all participants made accurate claims about their wins, ties, and losses leads to a contradiction. Therefore, at least one of the participants must have made a false claim.

Note: The specific participant who made the false claim cannot be determined based on the given information.