Olympiad problems

1963 All Russian Mathematical Olympiad, 028

Tags: combinatorics , Tournament

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?

1968 All Soviet Union Mathematical Olympiad, 108

Tags: combinatorics , Tournament

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?

1955 Moscow Mathematical Olympiad, 305

Tags: minimum , combinatorics , Tournament

$25$ 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?

2014 Contests, 3

Tags: combinatorics , Tournament

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 $7$.

1987 All Soviet Union Mathematical Olympiad, 441

Tags: Sum of Squares , algebra , combinatorics , Tournament

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 $$x_1^2 + ... + x_{10}^2 = y_1^2 + ... + y_{10}^2$$

2018 Bulgaria EGMO TST, 1

Tags: Tournament , combinatorics , results

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.

2024 ISI Entrance UGB, P8

Tags: combinatorics , Tournament , graph theory , ISI 2024

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 $1\le k \le N$, \[\frac{N - k}{2} \le x_k \le N - \frac{k+1}{2}\]

1969 All Soviet Union Mathematical Olympiad, 126

Tags: combinatorics , Tournament

$20$ 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]

2000 239 Open Mathematical Olympiad, 2

Tags: combinatorics , Tournament

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.

1973 All Soviet Union Mathematical Olympiad, 179

Tags: combinatorics , Tournament

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?

2015 USA Team Selection Test, 2

Tags: graph theory , combinatorics , Coloring , Tournament , Directed graphs , Hi

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]