PUMaC 2008 · 几何（B 组） · 第 5 题
PUMaC 2008 — Geometry (Division B) — Problem 5
题目详情
- (4 points) Two externally tangent circles have radius 2 and radius 3. Two lines are drawn, each tangent to both circles, but not at the point where the circles are tangent to each other. What is the area of the quadrilateral whose vertices are the four points of tangency between the circles and the lines?
解析
- A cube is divided into 27 unit cubes. A sphere is inscribed in each of the corner unit cubes, and another sphere is placed tangent to these 8 spheres. What is the smallest possible value for the radius of the last sphere? 1 Geometry √ 1 ( ANS: 3 − The smallest possible sphere is centered at the center of the cube, and each of the 2 8 corner spheres touches it where it is closest to the center. The center of each of the 8 spheres is √ 1 at a distance of 3 from the center of the cube, and the radius of each is . So, the point of 2 √ 1 tangency is at a distance of 3 − from the center of the cube, and that value must be the radius 2 of the last sphere. CB: ACH)