2019 EGMO Problems
Contents
[hide]Day 1
Problem 1
Find all triples of real numbers such that and
Problem 2
Let be a positive integer. Dominoes are placed on a board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each , determine the largest number of dominoes that can be placed in this way. (A domino is a tile of size or . Dominoes are placed on the board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. Two cells are said to be adjacent if they are different and share a common side.)
Problem 3
Let be a triangle such that , and let be its incentre. Let be the point on segment such that . Let be the circle tangent to at and passing through . Let be the second point of intersection of and the circumcircle of . Prove that the angle bisectors of and intersect at a point on line .
Day 2
Problem 4
Let be a triangle with incentre . The circle through tangent to at meets side again at . The circle through tangent to at meets side again at . Prove that is tangent to the incircle of
Problem 5
Let be an integer, and let be positive integers. Show that there exist positive integers satisfying the following three conditions:
(a) for
(b) the remainders of on division by are pairwise different; and
(c)
(Here, denotes the integer part of real number , that is, the largest integer that does not exceed .)
Problem 6
On a circle, Alina draws chords, the endpoints of which are all different. A point is considered marked if it is either
(i) one of the endpoints of a chord; or
(ii) an intersection point of at least two chords.
Alina labels each marked point. Of the points meeting criterion (i), Alina labels points with a and the other points with a . She labels each point meeting criterion (ii) with an arbitrary integer (not necessarily positive). Along each chord, Alina considers the segments connecting two consecutive marked points. (A chord with marked points has such segments.) She labels each such segment in yellow with the sum of the labels of its two endpoints and in blue with the absolute value of their difference. Alina finds that the yellow labels take each value exactly once. Show that at least one blue label is a multiple of . (A chord is a line segment joining two different points on a circle.)