2020 EGMO Problems
Contents
[hide]Day 1
Problem 1
The positive integers satisfy
Prove that at least one of the numbers
is divisible by
.
Problem 2
Find all lists of non-negative real numbers such that the following three conditions are all satisfied:
(i)
(ii)
(iii) there is a permutation of
such that
A permutation of a list is a list of the same length, with the same entries, but the entries are allowed to be in any order. For example, is a permutation of
, and they are both permutations of
. Note that any list is a permutation of itself.
Problem 3
Let be a convex hexagon such that
and
and the (interior) angle bisectors of
and
are concurrent.
Prove that the (interior) angle bisectors of and
must also be concurrent.
Note that . The other interior angles of the hexagon are similarly described.
Day 2
Problem 4
A permutation of the integers is called fresh if there exists no positive integer
such that the first
numbers in the permutation are
in some order. Let
be the number of fresh permutations of the integers
.
Prove that for all
.
For example, if , then the permutation
is fresh, whereas the permutation
is not.
Problem 5
Consider the triangle with
. The circumcircle
of
has radius
. There is a point
in the interior of the line segment
such that
and the length of
is
. The perpendicular bisector of
intersects
at the points
and
.
Prove is the incentre of triangle
.
Problem 6
Let be an integer. A sequence
is defined by
,
, and for all
,
Determine all integers
such that every term of the sequence is a square.