In a sports tournament involving $N$ teams, each team plays every other team exactly one. At the end of every match, the winning team gets $1$ point and losing team gets $0$ points. At the end of the tournament, the total points received by the individual teams are arranged in decreasing order as follows: \[x_1 \ge x_2 \ge \cdots \ge x_N . \] Prove that for any $1\le k \le N$, \[\frac{N - k}{2} \le x_k \le N - \frac{k+1}{2}\]

A tennis tournament has at least three participants. Every participant plays exactly one match against every other participant. Moreover, every participant wins at least one of the matches he plays. (Draws do not occur in tennis matches.) Show that there are three participants $A, B $ and $C$ for which the following holds: $A$ wins against $B, B$ wins against $C$, and $C$ wins against $A$.

$n\ge 2$ teams participated in an underwater polo tournament, each two teams played exactly once against each other. A team receives $2, 1, 0$ points for a win, draw, and loss correspondingly. It turned out that all teams got distinct numbers of points. In the final standings, the teams were ordered by the total number of points. A few days later, organizers realized that the results in the final standings were wrong due to technical issues: in fact, each match that ended with a draw according to them in fact had a winner, and each match with a winner in fact ended with a draw. It turned out that all teams still had distinct number of points! They corrected the standings and ordered them by the total number of points. For which $n$ could the correct order turn out to be the reversed initial order? [i](Proposed by Fedir Yudin)[/i]

In one country, a one-round tennis tournament was held (everyone played with everyone exactly once). Participants received $1$ point for winning a match, and $0$ points for losing. There are no draws in tennis. At the end of the tournament, Oleksiy saw the number of points scored by each participant, as well as the schedule of all the matches in the tournament, which showed the pairs of players, but not the winners. He chooses matches one by one in any order he wants and tries to guess the winner, after which he is told if he is correct. Prove that Oleksiy can act in such a way that he is guaranteed to guess the winners of more than half of the matches. [i]Proposed by Oleksiy Masalitin[/i]

$25$ chess players are going to participate in a chess tournament. All are on distinct skill levels, and of the two players the one who plays better always wins. What is the least number of games needed to select the two best players?

Six teams participate in a hockey tournament. Each team plays exactly once against each other team. A team is awarded $3$ points for each game they win, $1$ point for each draw, and $0$ points for each game they lose. After the tournament, a ranking is made. There are no ties in the list. Moreover, it turns out that each team (except the very last team) has exactly $2$ points more than the team ranking one place lower. Prove that the team that finished fourth won exactly two games.

$2022$ teams participated in an underwater polo tournament, each two teams played exactly once against each other. Team receives $2, 1, 0$ points for win, draw, and loss correspondingly. It turned out that all teams got distinct numbers of points. In the final standings the teams were ordered by the total number of points. A few days later, organizers realized that the results in the final standings were wrong due to technical issues: in fact, each match that ended with a draw according to them in fact had a winner, and each match with a winner in fact ended with a draw. It turned out that all teams still had distinct number of points! They corrected the standings, and ordered them by the total number of points. Could the correct order turn out to be the reversed initial order? [i](Proposed by Fedir Yudin)[/i]

Ten sportsmen have taken part in a table-tennis tournament (each pair has met once only, no draws). Let $xi$ be the number of $i$-th player victories, $yi$ -- losses. Prove that $$x_1^2 + ... + x_{10}^2 = y_1^2 + ... + y_{10}^2$$

$20$ football teams take part in a tournament . On the first day all the teams play one match . On the second day all the teams play a further match . Prove that after the second day it is possible to select $10$ teams, so that no two of them have yet played each other. ( S . A . Genkin)

A chess tournament took place between $2n+1$ players. Every player played every other player once, with no draws. In addition, each player had a numerical rating before the tournament began, with no two players having equal ratings. It turns out there were exactly $k$ games in which the lower-rated player beat the higher-rated player. Prove that there is some player who won no less than $n-\sqrt{2k}$ and no more than $n+\sqrt{2k}$ games.

$20$ football teams participate in the championship. What minimal number of the games should be played to provide the property: [i] from the three arbitrary teams we can find at least on pair that have already met in the championship.[/i]

$n$ players ($n \ge 4$) took part in the tournament. Each player played exactly one match with every other player, there were no draws. There was no four players $(A, B, C, D)$, such that $A$ won with $B$, $B$ won with $C$, $C$ won with $D$ and $D$ won with $A$. Determine, depending on $n$, maximum number of trios of players $(A, B, C)$, such that $A$ won with $B$, $B$ won with $C$ and $C$ won with $A$. (Attention: Trios $(A, B, C)$, $(B, C, A)$ and $(C, A, B)$ are the same trio.)