HMMT 二月 2015 · 团队赛 · 第 2 题

HMMT February 2015 — Team Round — Problem 2

[ 10 ] Let P be a (non-self-intersecting) polygon in the plane. Let C , . . . , C be circles in the plane 1 n whose interiors cover the interior of P . For 1 ≤ i ≤ n , let r be the radius of C . Prove that there is a i i single circle of radius r + · · · + r whose interior covers the interior of P . 1 n √

解析

[ 10 ] Let P be a (non-self-intersecting) polygon in the plane. Let C , . . . , C be circles in the plane 1 n whose interiors cover the interior of P . For 1 ≤ i ≤ n , let r be the radius of C . Prove that there is a i i single circle of radius r + · · · + r whose interior covers the interior of P . 1 n Answer: N/A If n = 1, we are done. Suppose n > 1. Since P is connected, there must be a point x on the plane which lies in the interiors of two circles, say C , C . Let O , O , respectively, be i j i j the centers of C , C . Since O O < r + r , we can choose O to be a point on segment O O such that i j i j i j i j O O ≤ r and O O ≤ r . Replace the two circles C and C with the circle C centered at O of radius i j j i i j r + r . Note that C covers both C and C . Induct to finish. i j i j √