HMMT 十一月 2014 · GEN 赛 · 第 1 题

HMMT November 2014 — GEN Round — Problem 1

Two circles ω and γ have radii 3 and 4 respectively, and their centers are 10 units apart. Let x be the shortest possible distance between a point on ω and a point on γ , and let y be the longest possible distance between a point on ω and a point on γ . Find the product xy . ◦

解析

Two circles ω and γ have radii 3 and 4 respectively, and their centers are 10 units apart. Let x be the shortest possible distance between a point on ω and a point on γ , and let y be the longest possible distance between a point on ω and a point on γ . Find the product xy . Answer: 51 Let ℓ be the line connecting the centers of ω and γ . Let A and B be the intersections of ℓ with ω , and let C and D be the intersections of ℓ with γ , so that A , B , C , and D are collinear, in that order. The shortest distance between a point on ω and a point on γ is BC = 3. The longest distance is AD = 3 + 10 + 4 = 17. The product is 51. ◦