MathDB

1

Graph colouring (6 vertices, two colours)

Six points are connected in pairs by lines, each of which is either red or blue. Every pair of points is joined. Determine whether there must be a closed path having four sides all of the same colour. (A path is closed if it begins and ends at the same point.)

5

1

Circle, point, gravity and time!

A circle and a point

P

higher than the circle lie in the same vertical plane. A particle moves along a straight line under gravity from

P

to a point

Q

on the circle. Given that the distance travelled from

P

in time

t

is equal to

\dfrac{1}{2}gt^2 \sin{\alpha}

, where

\alpha

is the angle of inclination of the line

PQ

to the horizontal, give a geometrical characterization of the point

Q

for which the time taken from

P

to

Q

is a minimum.

4

1

Functional equation with integers

Find all functions

f: \mathbb{Z} \rightarrow \mathbb{Z}

which satisfy

f(m + f(n)) = f(m) + n

for all

m,n \in \mathbb{Z}

.

2

1

The resulting numbers is 7/2 times the original one...

Find all natural numbers with the property that, when the first digit is moved to the end, the resulting number is

\dfrac{7}{2}

times the original one.

3

1

Diophantine equation with powers of 2 and 3

Find all solutions

x,y \in \mathbb{Z}

,

x,y \geq 0

, to the equation

1 + 3^x = 2^y.

1

Sequence of triangles

From an initial triangle

\Delta A_0B_0C_0

, a sequence of triangles

\Delta A_1B_1C_1

,

A_2B_2C_2

, ... is formed such that, at each stage,

A_{k + 1}

,

B_{k + 1}

and

C_{k + 1}

are the points where the incircle of

\Delta A_kB_kC_k

touches the sides

B_kC_k

,

C_kA_k

and

A_kB_k

respectively. (a) Express

\angle A_{k + 1}B_{k + 1}C_{k + 1}

in terms of

\angle A_kB_kC_k

. (b) Deduce that, as

k

increases,

\angle A_kB_kC_k

tends to

60^{\circ}

.

1997 South africa National Olympiad

Subcontests

Graph colouring (6 vertices, two colours)

Circle, point, gravity and time!

Functional equation with integers

The resulting numbers is 7/2 times the original one...

Diophantine equation with powers of 2 and 3

Sequence of triangles