# Canadian Mathematical Olympiad <br> 1990 

## PROBLEM 1

A competition involving $n \geq 2$ players was held over $k$ days. On each day, the players received scores of $1,2,3, \ldots, n$ points with no two players receiving the same score. At the end of the $k$ days, it was found that each player had exactly 26 points in total. Determine all pairs $(n, k)$ for which this is possible.

## Problem 2

A set of $\frac{1}{2} n(n+1)$ distinct numbers is arranged at random in a triangular array:
![](https://cdn.mathpix.com/cropped/2024_11_22_d44c6cbb61dbb677df78g-1.jpg?height=228&width=532&top_left_y=870&top_left_x=702)

Let $M_{k}$ be the largest number in the $k$-th row from the top. Find the probability that

$$
M_{1}<M_{2}<M_{3}<\cdots<M_{n}
$$

## Problem 3

Let $A B C D$ be a convex quadrilateral inscribed in a circle, and let diagonals $A C$ and $B D$ meet at $X$. The perpendiculars from $X$ meet the sides $A B, B C, C D, D A$ at $A^{\prime}, B^{\prime}, C^{\prime}, D^{\prime}$ respectively. Prove that

$$
\left|A^{\prime} B^{\prime}\right|+\left|C^{\prime} D^{\prime}\right|=\left|A^{\prime} D^{\prime}\right|+\left|B^{\prime} C^{\prime}\right|
$$

( $\left|A^{\prime} B^{\prime}\right|$ is the length of line segment $A^{\prime} B^{\prime}$, etc.)

## Problem 4

A particle can travel at speeds up to 2 metres per second along the $x$-axis, and up to 1 metre per second elsewhere in the plane. Provide a labelled sketch of the region which can be reached within one second by the particle starting at the origin.

## Problem 5

Suppose that a function $f$ defined on the positive integers satisfies

$$
\begin{gathered}
f(1)=1, \quad f(2)=2 \\
f(n+2)=f(n+2-f(n+1))+f(n+1-f(n)) \quad(n \geq 1)
\end{gathered}
$$

(a) Show that
(i) $0 \leq f(n+1)-f(n) \leq 1$
(ii) if $f(n)$ is odd, then $f(n+1)=f(n)+1$.
(b) Determine, with justification, all values of $n$ for which

$$
f(n)=2^{10}+1
$$