HMMT 二月 2023 · 冲刺赛 · 第 7 题

HMMT February 2023 — Guts Round — Problem 7

[11] Let Ω be a sphere of radius 4 and Γ be a sphere of radius 2. Suppose that the center of Γ lies on the surface of Ω. The intersection of the surfaces of Ω and Γ is a circle. Compute this circle’s circumference. √ √ √

解析

[11] Let Ω be a sphere of radius 4 and Γ be a sphere of radius 2. Suppose that the center of Γ lies on the surface of Ω. The intersection of the surfaces of Ω and Γ is a circle. Compute this circle’s circumference. Proposed by: Luke Robitaille √ Answer: π 15 Solution: Take a cross-section of a plane through the centers of Ω and Γ, call them O and O , 1 2 respectively. The resulting figure is two circles, one of radius 4 and center O , and the other with 1 radius 2 and center O on the circle of radius 4. Let these two circles intersect at A and B . Note that 2 AB is a diameter of the desired circle, so we will find AB . Focus on triangle O O A . The sides of this triangle are O O = O A = 4 and O A = 2. The height 1 2 1 2 1 2 √ √ √ 15 2 2 from O to AO is 4 − 1 = 15, and because O O = 2 · AO , the height from A to O O is . 1 2 1 2 2 1 2 2 √ Then the distance AB is two times this, or 15. √ Thus, the circumference of the desired circle is π 15. √ √ √