2021 EGMO Problems
Contents
[hide]Day 1
Problem 1
The number 2021 is fantabulous. For any positive integer , if any element of the set is fantabulous, then all the elements are fantabulous. Does it follow that the number is fantabulous?
Problem 2
Find all functions such that the equation holds for all rational numbers and .
Here, denotes the set of rational numbers.
Problem 3
Let be a triangle with an obtuse angle at . Let and be the intersections of the external bisector of angle with the altitudes of through and respectively. Let and be the points on the segments and respectively such that and . Prove that the points lie on a circle.
Day 2
Problem 4
Let be a triangle with incenter and let be an arbitrary point on the side . Let the line through perpendicular to intersect at . Let the line through perpendicular to intersect at . Prove that the reflection of across the line lies on the line .
Problem 5
A plane has a special point called the origin. Let be a set of 2021 points in the plane such that
(i) no three points in lie on a line and
(ii) no two points in lie on a line through the origin.
A triangle with vertices in is fat if is strictly inside the triangle. Find the maximum number of fat triangles.
Problem 6
Does there exist a nonnegative integer for which the equation has more than one million different solutions where and are positive integers?
The expression denotes the integer part (or floor) of the real number . Thus and .