HMMT 二月 2015 · 团队赛 · 第 2 题
HMMT February 2015 — Team Round — Problem 2
题目详情
英文原题
- [ 10 ] Let P be a (non-self-intersecting) polygon in the plane. Let C , . . . , C be circles in the plane
1 nwhose interiors cover the interior of P . For 1 ≤ i ≤ n , let r be the radius of C . Prove that there is ai isingle circle of radius r + · · · + r whose interior covers the interior of P .
1 n
√
解析
英文解析
- [ 10 ] Let P be a (non-self-intersecting) polygon in the plane. Let C , . . . , C be circles in the plane
1 nwhose interiors cover the interior of P . For 1 ≤ i ≤ n , let r be the radius of C . Prove that there is ai isingle 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 apoint x on the plane which lies in the interiors of two circles, say C , C . Let O , O , respectively, bei j i jthe centers of C , C . Since O O < r + r , we can choose O to be a point on segment O O such thati 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 radiusi j j i i jr + r . Note that C covers both C and C . Induct to finish.
i j i j
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