MathDB

3

Part of 2021 Middle European Mathematical Olympiad

Problems(2)

Easy Geometry

Source: 2021 MEMO I-3

9/5/2021
Let ABCABC be an acute triangle and DD an interior point of segment BCBC. Points EE and FF lie in the half-plane determined by the line BCBC containing AA such that DEDE is perpendicular to BEBE and DEDE is tangent to the circumcircle of ACDACD, while DFDF is perpendicular to CFCF and DFDF is tangent to the circumcircle of ABDABD. Prove that the points A,D,EA, D, E and FF are concyclic.
geometrycircumcircleConcyclicptolemy sinus lemmaComputer problems
Captain jack distribute coins equally

Source: 2021 MEMO T-3

9/5/2021
Let n,bn, b and cc be positive integers. A group of nn pirates wants to fairly split their treasure. The treasure consists of c⋅nc \cdot n identical coins distributed over b⋅nb \cdot n bags, of which at least n−1n-1 bags are initially empty. Captain Jack inspects the contents of each bag and then performs a sequence of moves. In one move, he can take any number of coins from a single bag and put them into one empty bag. Prove that no matter how the coins are initially distributed, Jack can perform at most n−1n-1 moves and then split the bags among the pirates such that each pirate gets bb bags and cc coins.
combinatoricsStrategyMEMO 2021memoinduction