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.