MathDB

2

Part of 2022 Rioplatense Mathematical Olympiad

Problems(4)

Rioplatense 2022 - Level 3 - Problem 2

Source:

12/6/2022
Let ABCABC be an acute triangle with AB<ACAB<AC. Let D,E,FD,E,F be the feet of the altitudes relatives to the vertices A,B,CA,B,C, respectively. The circumcircle Γ\Gamma of AEFAEF cuts the circumcircle of ABCABC at AA and MM. Assume that BMBM is tangent to Γ\Gamma. Prove that MM, FF and DD are collinear.
geometry
L+2x2=3x3

Source: Rioplatense L-2 2022 #2

12/13/2022
Let m,n≥2m,n\geq 2. One needs to cover the table m×nm \times n using only the following tiles: Tile 1 - A square 2×22 \times 2. Tile 2 - A L-shaped tile with five cells, in other words, the square 3×33 \times 3 without the upper right square 2×22 \times 2. Each tile 1 covers exactly 44 cells and each tile 2 covers exactly 55 cells. Rotation is allowed. Determine all pairs (m,n)(m,n), such that the covering is possible.

geometryrotationcombinatorics
Rioplatense 2022 - Level 1 - Problem 2

Source:

12/6/2022
Four teams AA, BB, CC and DD play a football tournament in which each team plays exactly two times against each of the remaining three teams (there are 1212 matches). In each matchif it's a tie each team gets 11 point and if it isn't a tie then the winner gets 33 points and the loser gets 00 points. At the end of the tournament the teams AA, BB and CC have 88 points each. Determine all possible points of team DD.
combinatorics
Rioplatense 2022 - Level A - Problem 2

Source:

12/6/2022
Eight teams play a rugby tournament in which each team plays exactly one match against each of the remaining seven teams. In each match, if it's a tie each team gets 11 point and if it isn't a tie then the winner gets 22 points and the loser gets 00 points. After the tournament it was observed that each of the eight teams had a different number of points and that the number of points of the winner of the tournament was equal to the sum of the number of points of the last four teams. Give an example of a tournament that satisfies this conditions, indicating the number of points obtained by each team and the result of each match.
combinatorics