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Determine the behavior of t_i for i→∞.

Source: IMO LongList 1970 - P32

May 21, 2011

functiongeometry unsolvedgeometry

Problem Statement

Let there be given an acute angle

\angle AOB = 3\alpha

, where

\overline{OA}= \overline{OB}

. The point

A

is the center of a circle with radius

\overline{OA}

. A line

s

parallel to

OA

passes through

B

. Inside the given angle a variable line

t

is drawn through

O

. It meets the circle in

O

and

C

and the given line

s

in

D

, where

\angle AOC = x

. Starting from an arbitrarily chosen position

t_0

of

t

, the series

t_0, t_1, t_2, \ldots

is determined by defining

\overline{BD_{i+1}}=\overline{OC_i}

for each

i

(in which

C_i

and

D_i

denote the positions of

C

and

D

, corresponding to

t_i

). Making use of the graphical representations of

BD

and

OC

as functions of

x

, determine the behavior of

t_i

for

i\to \infty

.

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