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.

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 $f_j(r) =\min (jr, n)+\min\left(\frac{j}{r}, n\right),\text{ and }g_j(r) =\min (\lceil jr\rceil, n)+\min\left(\left\lceil\frac{j}{r}\right\rceil, n\right),$ 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

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2013 Canadian MO Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5