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8 circles each tangent to 2, in an annulus between concentric circles

Source: Spanish Mathematical Olympiad 1973 P4

December 5, 2022

geometryconcentricconcentric circles

Problem Statement

Let

C

and

C'

be two concentric circles of radii

r

and

r'

respectively. Determine how much the quotient

r'/r

must be worth so that in the limited crown (annulus) through

C

and

C'

there are eight circles

C_i

i = 1, . . . , 8

, which are tangent to

C

and to

C'

, and also that

C_i

is tangent to

C_{i+1}

for

i = 1, . . . ,7

and

C_8

tangent to

C_1