2012 Pan African

Part of Pan African

Subcontests

(3)

2

Equation with Floor Functions

Find all real solutions

x

to the equation

\lfloor x^2 - 2x \rfloor + 2\lfloor x \rfloor = \lfloor x \rfloor^2

Median, altitude relationships

(i) Find the angles of

\triangle ABC

if the length of the altitude through

B

is equal to the length of the median through

C

and the length of the altitude through

C

is equal to the length of the median through

B

.
(ii) Find all possible values of

\angle ABC

\triangle ABC

if the length of the altitude through

A

is equal to the length of the median through

C

and the length of the altitude through

C

is equal to the length of the median through

B

2

Find all functions from reals to reals

Find all functions

f:\mathbb{R}\rightarrow\mathbb{R}

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

for all real numbers

x

y

n^m - m divides m^2 + 2m

Find all positive integers

m

n

n^m - m

m^2 + 2m

2

Circle, prove two products are equal

AB

is a chord (not a diameter) of a circle with centre

O

T

be a point on segment

OB

. The line through

T

perpendicular to

OB

AB

C

and the circle at

D

E

S

the orthogonal projection of

T

AB

AS \cdot BC = TE \cdot TD

Reciprocals on blackboard

\frac{1}{1}, \frac{1}{2}, \cdots , \frac{1}{2012}

are written on the blackboard. Aïcha chooses any two numbers from the blackboard, say

x

y

, erases them and she writes instead the number

x + y + xy

. She continues to do this until only one number is left on the board. What are the possible values of the final number?