MathDB

2

Part of 2017 European Mathematical Cup

Problems(2)

Sweet hexagons

Source: European mathematical cup 2017

1/3/2018
A regular hexagon in the plane is called sweet if its area is equal to 11. Is it possible to place 20000002000000 sweet hexagons in the plane such that the union of their interiors is a convex polygon of area at least 19000001900000?
Remark: A subset SS of the plane is called convex if for every pair of points in SS, every point on the straight line segment that joins the pair of points also belongs to SS. The hexagons may overlap.
combinatorics
Minimum number of players and substitution

Source: European Mathematical Cup 2017 Problem 2

12/27/2017
A friendly football match lasts 90 minutes. In this problem, we consider one of the teams, coached by Sir Alex, which plays with 11 players at all times.
a) Sir Alex wants for each of his players to play the same integer number of minutes, but each player has to play less than 60 minutes in total. What is the minimum number of players required? b) For the number of players found in a), what is the minimum number of substitutions required, so that each player plays the same number of minutes?
Remark: Substitutions can only take place after a positive integer number of minutes, and players who have come off earlier can return to the game as many times as needed. There is no limit to the number of substitutions allowed.
Proposed by Athanasios Kontogeorgis and Demetres Christofides.
combinatorics