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a sequence of circles, NT?

Source: Yugoslav TST 1978 P2

May 30, 2021
number theorygeometry

Problem Statement

Let k0k_0 be a unit semi-circle with diameter ABAB. Assume that k1k_1 is a circle of radius r1=12r_1=\frac12 that is tangent to both k0k_0 and ABAB. The circle kn+1k_{n+1} of radius rn+1r_{n+1} touches kn,k0k_n,k_0, and ABAB. Prove that:
(a) For each n∈{2,3,…}n\in\{2,3,\ldots\} it holds that 1rn+1+1rn−1=6rn−4\frac1{r_{n+1}}+\frac1{r_{n-1}}=\frac6{r_n}-4. (b) 1rn\frac1{r_n} is either a square of an even integer, or twice a square of an odd integer.
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