CANADA_MO/md/en-exam1970.md

Canadian Mathematical Olympiad

1970

Problem 1

Find all number triples $(x, y, z)$ such that when any one of these numbers is added to the product of the other two, the result is 2 .

Problem 2

Given a triangle $A B C$ with angle $A$ obtuse and with altitudes of length $h$ and $k$ as shown in the diagram, prove that $a+h \geq b+k$. Find under what conditions $a+h=b+k$.

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 2 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 this 6 is $1 / 25$ of the original integer. b) Show that there is no integer such that deletion of the first digit produces a result which 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

$$2\le a^2+b^2+c^2+d^2\le 42\; \backslash leq\; a^\{2\}+b^\{2\}+c^\{2\}+d^\{2\}\; \backslash leq\; 4$$

Problem 6

Given three non-collinear points $A, B, C$, construct a circle with centre $C$ such that the tangents from $A$ and $B$ to the circle 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 $y=2 x$. 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

$$0,1,1,2,2,3,3,4,4,5,5,6,6,\dots 0,1,1,2,2,3,3,4,4,5,5,6,6,\; \backslash ldots$$

a) Give a formula for $f(n)$. b) Prove that $f(s+t)-f(s-t)=s t$ where $s$ and $t$ are positive integers and $s>t$.

Problem 10 Given the polynomial

$$f(x)=x^n+a_1{x}^{n-1}+a_2{x}^{n-2}+\cdots +a_{n-1}x+a_{n}f(x)=x^\{n\}+a\_\{1\}\; x^\{n-1\}+a\_\{2\}\; x^\{n-2\}+\backslash cdots+a\_\{n-1\}\; x+a\_\{n\}$$

with integral coefficients $a_{1}, a_{2}, \ldots, a_{n}$, and given also that there exist four distinct integers $a, b, c$ and $d$ such that

$$f(a)=f(b)=f(c)=f(d)=5f(a)=f(b)=f(c)=f(d)=5$$

show that there is no integer $k$ such that $f(k)=8$.