2013 Canadian MO Problems

Problem 1

Determine all polynomials P(x) with real coefficients such that (x+1)P(x-1)-(x-1)P(x) is a constant polynomial.

Solution

Problem 2

The sequence a_1, a_2, \dots, a_n consists of the numbers 1, 2, \dots, n in some order. For which positive integers n is it possible that the n+1 numbers 0, a_1, a_1+a_2, a_1+a_2+a_3,\dots, a_1 + a_2 +\cdots + a_n all have di fferent remainders when divided by n + 1?

Solution

Problem 3

Let G be the centroid of a right-angled triangle ABC with \angle BCA = 90^\circ. Let P be the point on ray AG such that \angle CPA = \angle CAB, and let Q be the point on ray BG such that \angle CQB = \angle ABC. Prove that the circumcircles of triangles AQG and BPG meet at a point on side AB.

Solution

Problem 4

Let n be a positive integer. For any positive integer j and positive real number r, define f_j(r) and g_j(r) by f_j(r) = \min (jr, n) + \min\left(\frac{j}{r}, n\right), \text{ and } g_j(r) = \min (\lceil jr\rceil, n) + \min \left(\left\l... where \lceil x\rceil denotes the smallest integer greater than or equal to x. Prove that \sum_{j=1}^n f_j(r)\leq n^2+n\leq \sum_{j=1}^n g_j(r) for all positive real numbers r.

Solution

Problem 5

Let O denote the circumcentre of an acute-angled triangle ABC. Let point P on side AB be such that \angle BOP = \angle ABC, and let point Q on side AC be such that \angle COQ = \angle ACB. Prove that the reflection of BC in the line PQ is tangent to the circumcircle of triangle APQ.

Solution