DISINI exam1997

Canadian Mathematical Olympiad

1997

PROBLEM 1

How many pairs of positive integers x;y are there, with x y, and such that

gcd(x;y) = 5! and lcd(x;y) = 50!.

NOTE. gcd(x;y) denotes the greatest common divisor ofxandy, lcd(x;y) denotes

the least common multiple ofxandy, andn! =n(n,1)21.

PROBLEM 2

The closed intervalA = [0;50] is the union of a nite 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 length25.

NOTE. Forab, the closed interval [a;b] :=fx2R:axbghas lengthb,a;

disjoint intervals haveemptyintersection.

PROBLEM 3

Prove that ₁ 1999 <

1 2

3 4

5 6

1997 1998 <

1 44:

PROBLEM 4

The pointO is situated inside the parallelogramABCDso that

6

AOB+

6

COD= 180

:

Prove that6

OBC=

6

ODC. PROBLEM 5

Write the sum

n

X

k =0

(,1) k

,

n

k

k 3+ 9

k 2+ 26

k+ 24

in the form p(n) q (n), where

pandq are polynomials with integer coecients.