HMMT 二月 2016 · COMB 赛 · 第 1 题
HMMT February 2016 — COMB Round — Problem 1
题目详情
英文原题
- For positive integers n , let S be the set of integers x such that n distinct lines, no three concurrent,
can divide a plane into x regions (for example, S = { 3 , 4 } , because the plane is divided into 3 regionsnif the two lines are parallel, and 4 regions otherwise). What is the minimum i such that S contains at 2
least 4 elements?i
解析
英文解析
- For positive integers n , let S be the set of integers x such that n distinct lines, no three concurrent,
can divide a plane into x regions (for example, S = { 3 , 4 } , because the plane is divided into 3 regionsnif the two lines are parallel, and 4 regions otherwise). What is the minimum i such that S contains at 2
least 4 elements?i
Proposed by:
Answer: 4
For S , either all three lines are parallel (4 regions), exactly two are parallel (6 regions), or none areparallel (6 or seven regions, depending on whether they all meet at one point), so | S | = 3. Then, for 3
S , either all lines are parallel (5 regions), exactly three are parallel (8 regions), there are two sets of 3
parallel pairs (9 regions), exactly two are parallel (9 or 10 regions), or none are parallel (8, 9, 10, or 114
regions), so | S | = 4. 4