CANADA_MO/md/en-exam1990.md

Canadian Mathematical Olympiad
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:

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

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

$$\mid A^{\mathrm{\prime }}{B}^{\mathrm{\prime }}\mid +\mid C^{\mathrm{\prime }}{D}^{\mathrm{\prime }}\mid =\mid A^{\mathrm{\prime }}{D}^{\mathrm{\prime }}\mid +\mid B^{\mathrm{\prime }}{C}^{\mathrm{\prime }}\mid \backslash left|A^\{\backslash prime\}\; B^\{\backslash prime\}\backslash right|+\backslash left|C^\{\backslash prime\}\; D^\{\backslash prime\}\backslash right|=\backslash left|A^\{\backslash prime\}\; D^\{\backslash prime\}\backslash right|+\backslash left|B^\{\backslash prime\}\; C^\{\backslash prime\}\backslash right|$$

( $\left|A^{\prime} B^{\prime}\right|$ 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

$$\begin{array}{c}{\displaystyle f(1)=1,f(2)=2}\\ {\displaystyle f(n+2)=f(n+2-f(n+1))+f(n+1-f(n))(n\ge 1)}\end{array}\backslash begin\{gathered\}\; f(1)=1,\; \backslash quad\; f(2)=2\; \backslash \backslash \; f(n+2)=f(n+2-f(n+1))+f(n+1-f(n))\; \backslash quad(n\; \backslash geq\; 1)\; \backslash end\{gathered\}$$

(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

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