MathDB

1

Sequence of positive integers [Canada 1988 - P5]

S

is a sequence of positive integers let

p(S)

be the product of the members of

S

. Let

m(S)

be the arithmetic mean of

p(T)

for all non-empty subsets

T

of

S

. Suppose that

S'

is formed from

S

by appending an additional positive integer. If

m(S) = 13

and

m(S') = 49

, find

S'

.

4

1

Show that (b_n)^2 = 3(a_n)^ 2 + 1

Let

x_{n + 1} = 4x_n - x_{n - 1}

,

x_0 = 0

,

x_1 = 1

, and

y_{n + 1} = 4y_n - y_{n - 1}

,

y_0 = 1

,

y_1 = 2

. Show that for all

n \ge 0

that

y_n^2 = 3x_n^2 + 1

.

3

1

Show that we can find three points of the same color

Suppose that

S

is a finite set of at least five points in the plane; some are coloured red, the others are coloured blue. No subset of three or more similarly coloured points is collinear. Show that there is a triangle (i) whose vertices are all the same colour, and (ii) at least one side of the triangle does not contain a point of the opposite colour.

2

1

Find the area of S - [Canada 1988 - P2]

A house is in the shape of a triangle, perimeter

P

metres and area

A

square metres. The garden consists of all the land within 5 metres of the house. How much land do the garden and house together occupy?

1

For what values there is a common zero? [Canada 1988 - P1]

For what real values of

k

do

1988x^2 + kx + 8891

and

8891x^2 + kx + 1988

have a common zero?

1988 Canada National Olympiad

Subcontests

Sequence of positive integers [Canada 1988 - P5]

Show that (b_n)^2 = 3(a_n)^ 2 + 1

Show that we can find three points of the same color

Find the area of S - [Canada 1988 - P2]

For what values there is a common zero? [Canada 1988 - P1]