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sequence of circles, find radius

Source: Croatia MO 2001 4th Grade P1

April 22, 2021

geometrycircles

Problem Statement

On the unit circle

k_2

with center

O

, points

A

and

B

with

AB=1

are chosen and unit circles

k_1

and

k_2

with centers

A

and

B

are drawn. A sequence of circles

(l_n)

is defined as follows: circle

l_1

is tangent to

k

internally at

D_1

and to

k_1,k_2

externally, and for

n>1

circle

l_n

is tangent to

k_1

and

d_n=OD_n

and to

l_{n-1}

at

D_n

. For each

n

, compute

d_n=OD_n

and the radius

r_n

of

l_n

.

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