MathDB

2

If DEF is equilateral then AD, BE and CR are concurrent

P

lies inside a triangle

ABC

. Let

D,E

and

F

be reflections of the point

P

in the lines

BC,CA

and

AB

, respectively. Prove that if the triangle

DEF

is equilateral, then the lines

AD,BE

and

CF

intersect in a common point.

find all f such that f(x^2-y^2)=f(x)f(y)

Find all functions

f:\mathbb{N}_0\to\mathbb{N}_0

for which

f(0)=0

and

f(x^2-y^2)=f(x)f(y)

for all

x,y\in\mathbb{N}_0

with

x>y

.

1

2

Numbers in the set {-1,1} such that sum of pairs is 999

Determine whether or not there exist numbers

x_1,x_2,\ldots ,x_{2009}

from the set

\{-1,1\}

, such that:

x_1x_2+x_2x_3+x_3x_4+\ldots+x_{2008}x_{2009}+x_{2009}x_1=999

Minimum number of colours used for hats and ribbons

Consider

n

children in a playground, where

n\ge 2

. Every child has a coloured hat, and every pair of children is joined by a coloured ribbon. For every child, the colour of each ribbon held is different, and also different from the colour of that child’s hat. What is the minimum number of colours that needs to be used?

3

2

x must be an integer

Let

x

be a real number with the following property: for each positive integer

q

, there exists an integer

p

, such that

\left|x-\frac{p}{q} \right|<\frac{1}{3q}.

Prove that

x

is an integer.

Geometry Lots of Tangents and Chords

Points

C,E,D

and

F

lie on a circle with centre

O

. Two chords

CD

and

EF

intersect at a point

N

. The tangents at

C

and

D

intersect at

A

, and the tangents at

E

and

F

intersect at

B

. Prove that

ON\perp AB

.

2009 Pan African

Subcontests

If DEF is equilateral then AD, BE and CR are concurrent

find all f such that f(x^2-y^2)=f(x)f(y)

Numbers in the set {-1,1} such that sum of pairs is 999

Minimum number of colours used for hats and ribbons

x must be an integer

Geometry Lots of Tangents and Chords