MathDB

Problem 5

Part of 2022 Kyiv City MO Round 1

Problems(4)

Tournament went wrong

Source: Kyiv City MO 2022 Round 1, Problem 8.5

1/23/2022
20222022 teams participated in an underwater polo tournament, each two teams played exactly once against each other. Team receives 2,1,02, 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?
(Proposed by Fedir Yudin)
combinatoricsTournament
Tournament still broken

Source: Kyiv City MO 2022 Round 1, Problem 9.5

1/23/2022
n≥2n\ge 2 teams participated in an underwater polo tournament, each two teams played exactly once against each other. A team receives 2,1,02, 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 nn could the correct order turn out to be the reversed initial order?
(Proposed by Fedir Yudin)
Tournamentcombinatorics
Cute board game with tokens

Source: Kyiv City MO 2022 Round 1, Problem 10.5

1/23/2022
There is a black token in the lower-left corner of a board m×nm \times n (m,n≥3m, n \ge 3), and there are white tokens in the lower-right and upper-left corners of this board. Petryk and Vasyl are playing a game, with Petryk playing with a black token and Vasyl with white tokens. Petryk moves first.
In his move, a player can perform the following operation at most two times: choose any his token and move it to any adjacent by side cell, with one restriction: you can't move a token to a cell where at some point was one of the opponents' tokens.
Vasyl wins if at some point of the game white tokens are in the same cell. For which values of m,nm, n can Petryk prevent him from winning?
(Proposed by Arsenii Nikolaiev)
combinatoricsgame
Cutting square into squares of two sizes

Source: Kyiv City MO 2022 Round 1, Problem 11.5

1/23/2022
Find the smallest integer nn for which it's possible to cut a square into 2n2n squares of two sizes: nn squares of one size, and nn squares of another size.
(Proposed by Bogdan Rublov)
combinatoricscutting the paper