2005 Pan African

Part of Pan African

Subcontests

(3)

2

Want the line to half the area

ABC

be a triangle and let

P

be a point on one of the sides of

ABC

. Construct a line passing through

P

that divides triangle

ABC

into two parts of equal area.

Functional Equation on Z

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

be a function such that: For all

a

b

\mathbb{Z} - \{0\}

f(ab) \geq f(a) + f(b)

. Show that for all

a \in \mathbb{Z} - \{0\}

f(a^n) = nf(a)

n \in \mathbb{N}

f(a^2) = 2f(a)

2

Integer set and polynomial

S

be a set of integers with the property that any integer root of any non-zero polynomial with coefficients in

S

also belongs to

S

0

1000

are elements of

S

-2

is also an element of

S

Animal and Cage

Noah has to fit 8 species of animals into 4 cages of the Arc. He planes to put two species of animal in each cage. It turns out that, for each species of animal, there are at most 3 other species with which it cannot share a cage. Prove that there is a way to assign the animals to the cages so that each species shares a cage with a compatible species.

2

AM-HM like Inequality

For any positive real numbers

a,b

c

\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} \geq \dfrac{2}{a+b} + \dfrac{2}{b+c} + \dfrac{2}{c+a} \geq \dfrac{9}{a+b+c}

Greatest Integer Equation

[ {x} ]

be the greatest integer less than or equal to

x

\{x\}=x-[x]

. Solve the equation:

[x] \cdot \{x\} = 2005x