1970 Canadian MO Problems
Find all number triples such that when any of these numbers is added to the product of the other two, the result is 2.
Given a triangle with angle obtuse and with altitudes of length and as shown in the diagram, prove that . Find under what conditions .
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.
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.
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
Given three non-collinear points , construct a circle with centre such that the tangents from and are parallel.
Show that from any five integers, not necessarily distinct, one can always choose three of these integers whose sum is divisible by 3.
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.
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 .
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 .