1970 Canadian MO Problems
Contents
[hide]Problem 1
Find all number triples such that when any of these numbers is added to the product of the other two, the result is 2.
Problem 2
Given a triangle with angle
obtuse and with altitudes of length
and
as shown in the diagram, prove that
. Find under what conditions
.
Problem 3
A set of balls is given. Each ball is coloured red or blue, and there is at least one of each colour. Each ball weighs either 1 pound or 2 pounds, and there is at least one of each weight. Prove that there are two balls having different weights and different colours.
Problem 4
a) Find all positive integers with initial digit 6 such that the integer formed by deleting 6 is of the original integer.
b) Show that there is no integer such that the deletion of the first digit produces a result that is of the original integer.
Problem 5
A quadrilateral has one vertex on each side of a square of side-length 1. Show that the lengths ,
,
and
of the sides of the quadrilateral satisfy the inequalities
Problem 6
Given three non-collinear points , construct a circle with centre
such that the tangents from
and
are parallel.
Problem 7
Show that from any five integers, not necessarily distinct, one can always choose three of these integers whose sum is divisible by 3.
Problem 8
Consider all line segments of length 4 with one end-point on the line and the other end-point on the line
. Find the equation of the locus of the midpoints of these line segments.
Problem 9
Let be the sum of the first
terms of the sequence
a) Give a formula for
.
b) Prove that where
and
are positive integers and
.
Problem 10
Given the polynomial
with integer coefficients
, and given also that there exist four distinct integers
,
,
and
such that
show that there is no integer
such that
.