MathDB

Proof in a circle

Source: Cono Sur 1994-problem 2

June 1, 2006

geometry unsolvedgeometry

Problem Statement

Consider a circle

C

with diameter

AB=1

. A point

P_0

is chosen on

C

,

P_0 \ne A

, and starting in

P_0

a sequence of points

P_1, P_2, \dots, P_n, \dots

is constructed on

C

, in the following way:

Q_n

is the symmetrical point of

A

with respect of

P_n

and the straight line that joins

B

and

Q_n

cuts

C

at

B

and

P_{n+1}

(not necessary different). Prove that it is possible to choose

P_0

such that: i

\angle {P_0AB} < 1

. ii In the sequence that starts with

P_0

there are

2

points,

P_k

and

P_j

, such that

\triangle {AP_kP_j}

is equilateral.

Back to Problems View on AoPS