2015 EGMO Problems
Contents
[hide]Day 1
Problem 1
Let be an acute-angled triangle, and let
be the foot of the altitude from
The angle bisector of
intersects
at
and meets the circumcircle
of triangle
again at
If
, show that
is tangent to
Problem 2
A domino is a or
tile. Determine in how many ways exactly
dominoes can be placed without overlapping on a
chessboard so that every
square contains at least two uncovered unit squares which lie in the same row or column.
Problem 3
Let be integers greater than
, and let
be positive integers not greater than
. Prove that there exist positive integers
not greater than
, such that
where
denotes the greatest common divisor of
.
Day 2
Problem 4
Determine whether there exists an infinite sequence of positive integers
which satisfies the equality
for every positive integer
.
Problem 5
Let be positive integers with
. Anastasia partitions the integers
into
pairs. Boris then chooses one integer from each pair and finds the sum of these chosen integers.
Prove that Anastasia can select the pairs so that Boris cannot make his sum equal to
.
Problem 6
Let be the orthocentre and
be the centroid of acute-angled triangle
with
. The line
intersects the circumcircle of
at
and
. Let
be the reflection of
in the line
. Prove that
if and only if