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2022 LMT Spring Accuracy Problem 6

Source:

September 29, 2023
geometryLMT

Problem Statement

Jacob likes to watchMickeyMouse Clubhouse! One day, he decides to create his own MickeyMouse head shown below, with two circles ω1\omega_1 and ω2\omega_2 and a circle ω\omega, and centers O1O_1, O2O_2, and OO, respectively. Let ω1\omega_1 and ω\omega meet at points P1P_1 and Q1Q_1, and let ω2\omega_2 and ω\omega meet at points P2P_2 and Q2Q_2. Point P1P_1 is closer to O2O_2 than Q1Q_1, and point P2P_2 is closer to O1O_1 than Q2Q_2. Given that P1P_1 and P2P_2 lie on O1O2O_1O_2 such that O1P1=P1P2=P2O2=2O_1P_1 = P_1P_2 = P_2O_2 = 2, and Q1O1∥Q2O2Q_1O_1 \parallel Q_2O_2, the area of ω\omega can be written as nπn \pi. Find nn. https://cdn.artofproblemsolving.com/attachments/6/d/d98a05ee2218e80fd84d299d47201669736d99.png
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