1970 Canada National Olympiad

Part of Canada National Olympiad

Subcontests

(10)

1

Never 8

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

d

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

show that there is no integer

k

f(k)=8

1

Simple sequence

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

s

t

are positive integers and

s>t

1

Locus of midpoints

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.

1

Pick 3

Show that from any five integers, not necessarily distinct, one can always choose three of these integers whose sum is divisible by 3.

1

Construct a circle

Given three non-collinear points

A,B,C

, construct a circle with centre

C

such that the tangents from

A

B

1

Quadrilateral inside square

A quadrilateral has one vertex on each side of a square of side-length 1. Show that the lengths

a

b

c

d

of the sides of the quadrilateral satisfy the inequalities

2\le a^2+b^2+c^2+d^2\le 4.

1

Delete the digit!

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.

1

Coloured balls

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.

1

Obtuse triangle

Given a triangle

ABC

A

obtuse and with altitudes of length

h

k

as shown in the diagram, prove that

a+h\ge b+k

. Find under what conditions

a+h=b+k

. [asy] size(6cm);
pair A = dir(105), C = dir(170), B = dir(10), D = foot(B, A, C), E = foot(A, B, C);
draw(A--B--C--cycle); draw(B--D--A--E);
dot(A); dot(B); dot(C); dot(D); dot(E);
label("

A

", A, dir(110)); label("

B

", B, B); label("

C

", C, C); label("

D

", D, D); label("

E

", E, dir(45));
label("

h

", A--E, dir(0)); label("

k

", B--D, dir(45)); [/asy]

1

Find all triples

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.