# Canadian Mathematical Olympiad 

1997

## Problem 1

How many pairs of positive integers $x, y$ are there, with $x \leq y$, and such that $\operatorname{gcd}(x, y)=5!$ and $\operatorname{lcd}(x, y)=50$ !.
Note. $\operatorname{gcd}(x, y)$ denotes the greatest common divisor of $x$ and $y, \operatorname{lcd}(x, y)$ denotes the least common multiple of $x$ and $y$, and $n!=n \times(n-1) \times \cdots \times 2 \times 1$.

## Problem 2

The closed interval $A=[0,50]$ is the union of a finite number of closed intervals, each of length 1. Prove that some of the intervals can be removed so that those remaining are mutually disjoint and have total length $\geq 25$.
Note. For $a \leq b$, the closed interval $[a, b]:=\{x \in \mathbb{R}: a \leq x \leq b\}$ has length $b-a$; disjoint intervals have empty intersection.

## Problem 3

Prove that

$$
\frac{1}{1999}<\frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdots \frac{1997}{1998}<\frac{1}{44}
$$

## Problem 4

The point $O$ is situated inside the parallelogram $A B C D$ so that

$$
\angle A O B+\angle C O D=180^{\circ}
$$

Prove that $\angle O B C=\angle O D C$.
PROblem 5
Write the sum

$$
\sum_{k=0}^{n} \frac{(-1)^{k}\binom{n}{k}}{k^{3}+9 k^{2}+26 k+24}
$$

in the form $\frac{p(n)}{q(n)}$, where $p$ and $q$ are polynomials with integer coefficients.