# 2013 Canadian MO Problems

## 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 .