MathDB

5

Part of 2022 Rioplatense Mathematical Olympiad

Problems(4)

2 player game, k coins on intersections of n lines

Source: Rioplatense 2022 L3 p5

12/20/2022
Let n≥4n \ge 4 and kk be positive integers. We consider nn lines in the plane between which there are not two parallel nor three concurrent. In each of the n(n−1)2\frac{n(n-1)}{2} points of intersection of these lines, kk coins are placed. Ana and Beto play the following game in turns: each player, in turn, chooses one of those points that does not share one of the nn lines with the point chosen immediately before by the other player, and removes a coin from that point. Ana starts and can choose any point. The player who cannot make his move loses. Determine based on nn and kk who has a winning strategy.
combinatoricsgame strategywinning strategygame
couples + numbers = perfect...

Source: Rioplatense L-2 2022 #5

12/13/2022
Let nn be a positive integer. The numbers 1,2,3,…,4n1,2,3,\dots, 4n are written in a board. Olive wants to make some "couples" of numbers, such that the product of the numbers in each couple is a perfect square. Each number is in, at most, one couple and the two numbers in each couple are distincts. Determine, for each positive integer nn, the maximum number of couples that Olive can write.
number theorycombinatorics
Rioplatense L-1 2022 #5

Source:

12/13/2022
Let ABCDEFGHIABCDEFGHI be a regular polygon with 99 sides and the vertices are written in the counterclockwise and let ABJKLMABJKLM be a regular polygon with 66 sides and the vertices are written in the clockwise. Prove that ∠HMG=∠KEL\angle HMG=\angle KEL. Note: The polygon ABJKLMABJKLM is inside of ABCDEFGHIABCDEFGHI.
geometry
Rioplatense L-A 2022 #5

Source:

12/13/2022
The quadrilateral ABCDABCD has the following equality ∠ABC=∠BCD=150∘\angle ABC=\angle BCD=150^{\circ}. Moreover, AB=18AB=18 and BC=24BC=24, the equilateral triangles △APB,△BQC,△CRD\triangle APB,\triangle BQC,\triangle CRD are drawn outside the quadrilateral. If P(X)P(X) is the perimeter of the polygon XX, then the following equality is true P(APQRD)=P(ABCD)+32P(APQRD)=P(ABCD)+32. Determine the length of the side CDCD.
geometryperimeter