1

Part of 2012 Pan African

Problems(2)

Circle, prove two products are equal

Source: Pan African MO 2012 Day 1 Q1

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

geometry proposedgeometry

Reciprocals on blackboard

Source: Pan African MO 2012 Day 2 Q1

\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?

inductionSFFTgeometry3D geometryinvariantspecial factorizationscombinatorics proposed