HMMT 二月 2016 · COMB 赛 · 第 1 题

HMMT February 2016 — COMB Round — Problem 1

For positive integers n , let S be the set of integers x such that n distinct lines, no three concurrent, n can divide a plane into x regions (for example, S = { 3 , 4 } , because the plane is divided into 3 regions 2 if the two lines are parallel, and 4 regions otherwise). What is the minimum i such that S contains at i least 4 elements?

解析

For positive integers n , let S be the set of integers x such that n distinct lines, no three concurrent, n can divide a plane into x regions (for example, S = { 3 , 4 } , because the plane is divided into 3 regions 2 if the two lines are parallel, and 4 regions otherwise). What is the minimum i such that S contains at i least 4 elements? Proposed by: Answer: 4 For S , either all three lines are parallel (4 regions), exactly two are parallel (6 regions), or none are 3 parallel (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 4 parallel pairs (9 regions), exactly two are parallel (9 or 10 regions), or none are parallel (8, 9, 10, or 11 regions), so | S | = 4. 4