Subcontests
(5)Classifying roads
m,n are positive integers such that m≥2, n<23(m−1). In a country there are m cities and n roads, each road connect two different cities, and there can be multiple roads between two cities. Prove that there exist a way to separate the cities into two groups α and β, where all roads connecting a city in α to a city in β is converted to a highway, and satisfies the following conditions:
[*]Both groups have at least one city, and
[*]for each city, the number of highways coming out from that city does not exceed 1. 2016 Japan Mathematical Olympiad Finals, Problem 2
Let ABCD be a concyclic quadrilateral such that AB:AD=CD:CB. The line AD intersects the line BC at X, and the line AB intersects the line CD at Y. Let E, F, G and H are the midpoints of the edges AB, BC, CD and DA respectively. The bisector of angle AXB intersects the segment EG at S, and that of angle AYD intersects the segment FH at T. Prove that the lines ST and BD are pararell.