2002 Pan African

Part of Pan African

Subcontests

(6)

1

Inequality with sum 1

a_1 \geq a_2 \geq \cdots \geq a_n \geq 0

a_1+a_2+\cdots+a_n=1

a_1^2+3a_2^2+5a_3^2+ \cdots +(2n-1)a_n^2 \leq 1

1

P on the altitude.

\triangle{ABC}

be an acute angled triangle. The circle with diameter AB intersects the sides AC and BC at points E and F respectively. The tangents drawn to the circle through E and F intersect at P. Show that P lies on the altitude through the vertex C.

1

Marks in exam

Seven students in a class compare their marks in 12 subjects studied and observe that no two of the students have identical marks in all 12 subjects. Prove that we can choose 6 subjects such that any two of the students have different marks in at least one of these subjects.

1

Write with 1 and 2

Prove for every integer

n>0

, there exists an integer

k>0

2^nk

can be written in decimal notation using only digits 1 and 2.

1

Perpendicular bisector passes through a point

\triangle{AOB}

is a right triangle with

\angle{AOB}=90^{o}

C

D

AO

BO

respectively such that

AC=BD

. Show that there is a fixed point

P

through which the perpendicular bisector of

CD

1

f(f(n))=f(n)+1 with minimum 1

Find all functions

f: N_0 \to N_0

N_0

is the set of all non-negative integers) such that

f(f(n))=f(n)+1

n \in N_0

and the minimum of the set

\{ f(0), f(1), f(2) \cdots \}

1