2017 EGMO Problems
Contents
[hide]Day 1
Problem 1
Let be a convex quadrilateral with and . Let and be points on segments and , respectively, such that line intersects lines and at points and , respectively. It is given that .Let the midpoint of be and the midpoint of be .Prove that the points and lie on a circle.
Problem 2
Find the smallest positive integer for which there exists a colouring of the positive integers with colours and a function with the following two properties:
(i) For all positive integers of the same colour,
(ii) There are positive integers such that
In a colouring of with colours, every integer is coloured in exactly one of the colours. In both (i) and (ii) the positive integers are not necessarily distinct.
Problem 3
There are 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?
Day 2
Problem 4
Let be an integer and let be positive integers. In a group of 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 .
(ii) For every with , there is someone who has played exactly games of chess.
Problem 5
Let be an integer. An -tuple of not necessarily different positive integers is expensive if there exists a positive integer such that (a) Find all integers for which there exists an expensive -tuple.
(b) Prove that for every odd positive integer there exists an integer such that belongs to an expensive -tuple.
There are exactly factors in the product on the left hand side.
Problem 6
Let be an acute-angled triangle in which no two sides have the same length. The reflections of the centroid and the circumcentre of in its sides are denoted by and , respectively. Show that the circumcircles of triangles , , , , , and have a common point.
The centroid of a triangle is the intersection point of the three medians. A median is a line connecting a vertex of the triangle to the midpoint of the opposite side.