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
.