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.

a) Two seventh graders and several eightth graders take part in a chess tournament. The two seventh graders together scored eight points. The scores of eightth graders are equal. How many eightth graders took part in the tournament? b) Ninth and tenth graders participated in a chess tournament. There were ten times as many tenth graders as ninth graders. The total score of tenth graders was $4.5$ times that of the ninth graders. What was the ninth graders score? Note: According to the rules of a chess tournament, each of the tournament participants ra plays one game with each of them. If one of the players wins the game, then he gets one point, and his opponent gets zero points. In case of a tie, the players receive 1/2 point.

In a chess tournament , every two participants play each other exactly once. A win is worth one point , a draw is worth half a point and a loss is worth zero points. Looking back at the end of the tournament, a game is called an upset if the total number of points obtained by the winner of that game is less than the total number of points obtained by the loser of that game. (a) Prove that the number of upsets is always strictly less than three-quarters of the total number of games in the tournament. (b) Prove that three-quarters cannot be replaced by a smaller number. (S Tokarev) PS. part (a) for Juniors, both parts for Seniors

In a qualification football round there are six teams and each two play one versus another exactly once. No two matches are played at the same time. At every moment the difference between the number of already played matches for any two teams is $0$ or $1$. A win is worth $3$ points, a draw is worth $1$ point and a loss is worth $0$ points. Determine the smallest positive integer $n$ for which it is possible that after the $n$-th match all teams have a different number of points and each team has a non-zero number of points.

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.

In the spring round of the Tournament of Towns this year, $6$ problems were posed in the Senior A-Level paper. In a certain country, each problem was solved by exactly $1000$ participants, but no two participants solved all $6$ problems between them. What is the smallest possible number of participants from this country in the spring round Senior A-Level paper? (R Zhenodarov)

In a football tournament of one round (each team plays each other once, $2$ points for win , $1$ point for draw and $0$ points for loss), $28$ teams compete. During the tournament more than $75\%$ of the matches finished in a draw . Prove that there were two teams who finished with the same number of points. (M . Vora, gymnasium student , Hungary)

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 $7$.

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.)

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) $k=3$.

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.

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?

Eight football teams participate in a tournament of one round (each team plays each other team once) . There are no draws. Prove that it is possible at the conclusion of the tournament to be able to find $4$ teams , say $A, B, C$ and $D$ so that $A$ defeated $B, C$ and $D, B$ defeated $C$ and $D$ , and $C$ defeated $D$ .

In the Faculty of Cybernetics football championship, \( n \geq 3 \) teams participated. The competition was held in a round-robin format, meaning that each team played against every other team exactly once. For a win, a team earns 3 points, for a loss no points are awarded, and for a draw, both teams receive 1 point each. It turned out that the winning team scored strictly more points than any other team and had at most as many wins as losses. What is the smallest \( n \) for which this could happen? [i]Proposed by Bogdan Rublov[/i]

The world and the european champion are determined in the same tournament carried in one round. There are $20$ teams and $k$ of them are european. The european champion is determined according to the results of the games only between those $k$ teams. What is the greatest $k$ such that the situation, when the single european champion is the single world outsider, is possible if: a) it is hockey (draws allowed)? b) it is volleyball (no draws)?

100 volleyball teams played a one-round tournament. No two matches held at the same time. It turned out that before each match the teams playing against each other had the same number of wins. Find all possible number of wins for the winner of this tournament.

One-round hockey tournament is finished (each plays with each one time, the winner gets $2$ points, looser -- $0$, and $1$ point for draw). For arbitrary subgroup of teams there exists a team (may be from that subgroup) that has got an odd number of points in the games with the teams of the subgroup. Prove that there was even number of the participants.

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.

In a basketball competition, $n$ teams took part. Each pair of teams played exactly one match, and there were no draws. At the end of the competition the $i$-th team had $x_i$ wins and $y_i$ defeats $(i=1,\ldots,n)$. Prove that $x_1^2+x_2^2+\ldots+x_n^2=y_1^2+y_2^2+\ldots+y_n^2$.

Eight men had participated in the chess tournament. (Each meets each, draws are allowed, giving $1/2$ of point, winner gets $1$.) Everyone has different number of points. The second one has got as many points as the four weakest participants together. What was the result of the play between the third prizer and the chess-player that have occupied the seventh place?

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]

Each of the $9$ referees on the figure skating championship estimates the program of $20$ sportsmen by assigning him a place (from $1$ to $20$). The winner is determined by adding those numbers. (The less is the sum - the higher is the final place). It was found, that for the each sportsman, the difference of the places, received from the different referees was not greater than $3$. What can be the maximal sum for the winner?

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 $7$.