1970 Canadian MO Problems

Problem 1

Find all number triples $(x,y,z)$ 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 $ABC$ with angle $A$ obtuse and with altitudes of length $h$ and $k$ as shown in the diagram, prove that $a+h\ge b+k$. Find under what conditions $a+h=b+k$.

[asy] draw((0,0)--(5,0)--(16/5,12/5)--cycle,dot); draw((2.5,0)--(2.5,7.5/4)--(5,0)--cycle,black); MP("C",(0,0),SW);MP("D",(16/5,12/5),N);MP("B",(5,0),SE); MP("E",(2.5,0),NE);MP("A",(2.5,7.5/4),N); MP("h",(2.5,7.5/8),W);MP("k",(41/10,6/5),NE); draw((-.2,.2)--(2.5-.2,7.5/4+.2),arrow=ArcArrow(TeXHead)); draw((2.5-.2,7.5/4+.2)--(-.2,.2),arrow=ArcArrow(TeXHead)); MP("b",(2.3/2-.05,7.5/8+.25),N); draw((0,-.2)--(5,-.2),arrow=ArcArrow(TeXHead)); draw((5,-.2)--(0,-.2),arrow=ArcArrow(TeXHead)); MP("a",(2.5,-.2),S); draw((16/5,12/5)--(16/5-.2,12/5-.15)--(16/5-.2+.15,12/5-.15-.2)--(16/5+.15,12/5-.2)--cycle,black); [/asy]


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 $1/25$ of the original integer.

b) Show that there is no integer such that the deletion of the first digit produces a result that is $1/35$ 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 $a$, $b$, $c$ and $d$ of the sides of the quadrilateral satisfy the inequalities $2\le a^2+b^2+c^2+d^2\le 4.$


Problem 6

Given three non-collinear points $A,B,C$, construct a circle with centre $C$ such that the tangents from $A$ and $B$ 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 $y=x$ and the other end-point on the line $y=2x$. Find the equation of the locus of the midpoints of these line segments.


Problem 9

Let $f(n)$ be the sum of the first $n$ terms of the sequence \[0, 1,1, 2,2, 3,3, 4,4, 5,5, 6,6, \ldots\, .\] a) Give a formula for $f(n)$.

b) Prove that $f(s+t)-f(s-t)=st$ where $s$ and $t$ are positive integers and $s>t$.


Problem 10

Given the polynomial \[f(x)=x^n+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-1}x+a_n\] with integer coefficients $a_1,a_2,\ldots,a_n$, and given also that there exist four distinct integers $a$, $b$, $c$ and $d$ such that \[f(a)=f(b)=f(c)=f(d)=5,\] show that there is no integer $k$ such that $f(k)=8$.


See Also