Five men play several sets of dominoes (two against two) so that each player has each other player once as a partner and two times as an opponent. Find the number of sets and all ways to arrange the players.

For each natural number $k$ find the least number $n$ such that in every tournament with $n$ vertices, there exists a vertex with in-degree and out-degree at least $k$. (Tournament is directed complete graph.)

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.

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.

On football toornament there were $4$ teams participating. Every team played exactly one match with every other team. For the win, winner gets $3$ points, while if draw both teams get $1$ point. If at the end of tournament every team had different number of points and first place team had $6$ points, find the points of other teams

At a volleyball tournament, each team plays exactly once against each other team. Each game has a winning team, which gets $1$ point. The losing team gets $0$ points. Draws do not occur. In the nal ranking, only one team turns out to have the least number of points (so there is no shared last place). Moreover, each team, except for the team having the least number of points, lost exactly one game against a team that got less points in the final ranking. a) Prove that the number of teams cannot be equal to $6$. b) Show, by providing an example, that the number of teams could be equal to $7$.

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 tournament is a directed graph for which every (unordered) pair of vertices has a single directed edge from one vertex to the other. Let us define a proper directed-edge-coloring to be an assignment of a color to every (directed) edge, so that for every pair of directed edges $\overrightarrow{uv}$ and $\overrightarrow{vw}$, those two edges are in different colors. Note that it is permissible for $\overrightarrow{uv}$ and $\overrightarrow{uw}$ to be the same color. The directed-edge-chromatic-number of a tournament is defined to be the minimum total number of colors that can be used in order to create a proper directed-edge-coloring. For each $n$, determine the minimum directed-edge-chromatic-number over all tournaments on $n$ vertices. [i]Proposed by Po-Shen Loh[/i]

At a volleyball tournament, each team plays exactly once against each other team. Each game has a winning team, which gets $1$ point. The losing team gets $0$ points. Draws do not occur. In the nal ranking, only one team turns out to have the least number of points (so there is no shared last place). Moreover, each team, except for the team having the least number of points, lost exactly one game against a team that got less points in the final ranking. a) Prove that the number of teams cannot be equal to $6$. b) Show, by providing an example, that the number of teams could be equal to $7$.

$n$ 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.)

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.

The tennis federation has assigned numbers to $1024$ sportsmen, participating in the tournament, according to their skill. (The tennis federation uses the olympic system of tournaments. The looser in the pair leaves, the winner meets with the winner of another pair. Thus, in the second tour remains $512$ participants, in the third -- $256$, et.c. The winner is determined after the tenth tour.) It comes out, that in the play between the sportsmen whose numbers differ more than on $2$ always win that whose number is less. What is the greatest possible number of the winner?

For a competition a school wants to nominate a team of $k$ students, where $k$ is a given positive integer. Each member of the team has to compete in the three disciplines juggling, singing and mental arithmetic. To qualify for the team, the $n \ge 2$ students of the school compete in qualifying competitions, determining a unique ranking in each of the three disciplines. The school now wants to nominate a team satisfying the following condition: $(*)$ [i]If a student $X$ is not nominated for the team, there is a student $Y$ on the team who defeated $X$ in at least two disciplines.[/i] Determine all positive integers $n \ge 2$ such that for any combination of rankings, a team can be chosen to satisfy the condition $(*)$, when a) $k=2$, b) $k=3$.

At the end of a soccer tournament in which any pair of teams played between them exactly once, and in which there were not draws, it was observed that for any three teams $A, B$ and C, if $A$ defeated $B$ and $B$ defeated $C$, then $A$ defeated $C$. Any team calculated the difference (positive) between the number of games that it won and the number of games it lost. The sum of all these differences was $5000$. How many teams played in the tournament? Find all possible answers.

Eighteen soccer teams have played $8$ tours of a one-round tournament. Prove that there is a triple of teams, having not met each other yet.

$20$ 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]

$25$ 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?