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Math Prize 2022 Problem 19

Source:

October 12, 2022

Problem Statement

Let S−S_- be the semicircular arc defined by (x+1)2+(y−32)2=14 and x≤−1. (x + 1)^2 + (y - \frac{3}{2})^2 = \frac{1}{4} \text{ and } x \le -1. Let S+S_+ be the semicircular arc defined by (x−1)2+(y−32)2=14 and x≥1. (x - 1)^2 + (y - \frac{3}{2})^2 = \frac{1}{4} \text{ and } x \ge 1. Let RR be the locus of points PP such that PP is the intersection of two lines, one of the form Ax+By=1Ax + By = 1 where (A,B)∈S−(A, B) \in S_- and the other of the form Cx+Dy=1Cx + Dy = 1 where (C,D)∈S+(C, D) \in S_+. What is the area of RR?
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