MathDB

2

Part of 2016 European Mathematical Cup

Problems(2)

European Mathematical Cup 2016 senior division problem 2

Source:

12/31/2016
For two positive integers aa and bb, Ivica and Marica play the following game: Given two piles of aa and bb cookies, on each turn a player takes 2n2n cookies from one of the piles, of which he eats nn and puts nn of them on the other pile. Number nn is arbitrary in every move. Players take turns alternatively, with Ivica going first. The player who cannot make a move, loses. Assuming both players play perfectly, determine all pairs of numbers (a,b)(a, b) for which Marica has a winning strategy.
Proposed by Petar Orlić
combinatorics
European Mathematical Cup 2016 junior division problem 2

Source:

12/31/2016
Two circles C1C_{1} and C2C_{2} intersect at points AA and BB. Let PP, QQ be points on circles C1C_{1}, C2C_{2} respectively, such that ∣AP∣=∣AQ∣|AP| = |AQ|. The segment PQPQ intersects circles C1C_{1} and C2C_{2} in points MM, NN respectively. Let CC be the center of the arc BPBP of C1C_{1} which does not contain point AA and let DD be the center of arc BQBQ of C2C_{2} which does not contain point AA Let EE be the intersection of CMCM and DNDN. Prove that AEAE is perpendicular to CDCD.
Proposed by Steve Dinh
geometryemc