Found problems: 36
1975 All Soviet Union Mathematical Olympiad, 218
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)?
2017 Dutch Mathematical Olympiad, 3
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.
2012 Polish MO Finals, 4
$n$ 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.)
2014 BAMO, 5
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.
1981 All Soviet Union Mathematical Olympiad, 317
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.
2000 Tournament Of Towns, 6
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)
1972 All Soviet Union Mathematical Olympiad, 173
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.
2023 Indonesia TST, C
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.
2009 Dutch Mathematical Olympiad, 3
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 $A$.
1986 Tournament Of Towns, (111) 5
$20$ 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)
2014 Dutch Mathematical Olympiad, 3
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 $7$.
2022 Iran MO (3rd Round), 1
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.)
1955 Moscow Mathematical Olympiad, 315
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.
2000 Tournament Of Towns, 6
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
1985 Tournament Of Towns, (097) 1
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 $D$ .
2012 Bosnia And Herzegovina - Regional Olympiad, 2
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
2023 German National Olympiad, 3
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) $k=3$.
2004 Mexico National Olympiad, 4
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.
2023 Indonesia TST, C
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.
1999 Croatia National Olympiad, Problem 4
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 $x_1^2+x_2^2+\ldots+x_n^2=y_1^2+y_2^2+\ldots+y_n^2$.
2025 Kyiv City MO Round 1, Problem 3
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]
2022 Kyiv City MO Round 1, Problem 5
$2022$ 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]
2022 Kyiv City MO Round 1, Problem 5
$n\ge 2$ 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]
2023 Ukraine National Mathematical Olympiad, 8.2
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]
1946 Moscow Mathematical Olympiad, 116
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.