2013 Canadian MO Problems
Contents
[hide]Problem 1
Determine all polynomials with real coefficients such that
is a constant polynomial.
Problem 2
The sequence consists of the numbers
in some order. For which positive integers
is it possible that the
numbers
all have di fferent remainders when divided by
?
Problem 3
Let be the centroid of a right-angled triangle
with
. Let
be the point on ray
such that
, and let
be the point on ray
such that
. Prove that the circumcircles of triangles
and
meet at a point on side
.
Problem 4
Let be a positive integer. For any positive integer
and positive real number
, define
where
denotes the smallest integer greater than or equal to
. Prove that
for all positive real numbers
.
Problem 5
Let denote the circumcentre of an acute-angled triangle
. Let point
on side
be such that
, and let point
on side
be such that
. Prove that the reflection of
in the line
is tangent to the circumcircle of triangle
.