MathDB

3

Part of 2017 EGMO

Problems(2)

Lines in Plane

Source: EGMO 2017 Day 1 P3

4/8/2017
There are 20172017 lines in the plane such that no three of them go through the same point. Turbo the snail sits on a point on exactly one of the lines and starts sliding along the lines in the following fashion: she moves on a given line until she reaches an intersection of two lines. At the intersection, she follows her journey on the other line turning left or right, alternating her choice at each intersection point she reaches. She can only change direction at an intersection point. Can there exist a line segment through which she passes in both directions during her journey?
PlaneLinecombinatoricsEGMOcombinatorial geometryEGMO 2017
Constructing graphs satisfying conditions on degrees

Source: EGMO 2017 P4

4/9/2017
Let n≥1n\geq1 be an integer and let t1<t2<⋯<tnt_1<t_2<\dots<t_n be positive integers. In a group of tn+1t_n+1 people, some games of chess are played. Two people can play each other at most once. Prove that it is possible for the following two conditions to hold at the same time:
(i) The number of games played by each person is one of t1,t2,…,tnt_1,t_2,\dots,t_n.
(ii) For every ii with 1≤i≤n1\leq i\leq n, there is someone who has played exactly tit_i games of chess.
combinatoricsgraph theoryvertex degreeEGMOEGMO 2017