MathDB

A circle

K_1

of radius r_1 = 1\slash 2 is inscribed in a semi-circle

H

with diameter

AB

and radius

1.

A sequence of different circles

K_2, K_3, \ldots

with radii

r_2, r_3, \ldots

respectively are drawn so that for each

n\geq 1

, the circle

K_{n+1}

is tangent to

H

K_n

and

AB.

Prove that a_n = 1\slash r_n is an integer for each

n

, and that it is a perfect square for

n

even and twice a perfect square for

n

odd.

Problem Statement