Problem:
A collection of circles in the upper half-plane, all tangent to the x-axis, is constructed in layers as follows. Layer L0​ consists of two circles of radii 702 and 732 that are externally tangent. For k≥1, the circles in ⋃j=0k−1​Lj​ are ordered according to their points of tangency with the x-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Laver Lk​ consists of the 2k−1 circles constructed in this way. Let S=⋃j=06​Lj​, and for every circle C denote by r(C) its radius. What is
C∈S∑​r(C)​1​?
Answer Choices:
A. 35286​
B. 70583​
C. 73715​
D. 14143​
E. 1461573​ Solution:
Suppose that circles C1​ and C2​ in the upper half-plane have
centers O1​ and O2​ and radii r1​ and r2​, respectively. Assume that C1​ and C2​ are externally tangent and tangent to the x-axis at X1​ and X2​, respectively. Let C with center O and radius r be the circle externally tangent to C1​ and C2​ and tangent to the x-axis. Let X be the point of tangency of C with the x-axis, and let T1​ and T2​ be the points of tangency of C with C1​ and C2​, respectively. Let M1​ and M2​ be the points on the x-axis such that M1​T1​​⊥O1​T1​​ and M2​T2​​⊥O2​T2​​.
Because M1​X1​​ and M1​T1​​ are both tangent to C1​, it follows that X1​M1​=M1​T1​. Similarly, M1​T1​​ and M1​X​ are both tangent to C, and thus M1​T1​=M1​X. Because ∠OT1​M1​,∠M1​X1​O1​,∠M1​T1​O, and ∠OXM1​ are all right angles and ∠T1​M1​X=π−∠X1​M1​T1​, it follows that quadrilaterals O1​X1​M1​T1​ and M1​XOT1​ are similar. Thus
where C1​ and C2​ are the consecutive circles in ⋃j=0k−1​Lj​ that are tangent to C. Note that every circle in ⋃j=0k−1​Lj​ appears twice in the sum on the right-hand side, except for the two circles in L0​, which appear only once. Thus
