HMMT 十一月 2024 · 冲刺赛 · 第 3 题

HMMT November 2024 — Guts Round — Problem 3

[5] The graphs of the lines y = x + 2 , y = 3 x + 4 , y = 5 x + 6 , y = 7 x + 8 , y = 9 x + 10 , y = 11 x + 12 are drawn. These six lines divide the plane into several regions. Compute the number of regions the plane is divided into. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2024, November 09, 2024 — GUTS ROUND Organization Team Team ID# 6

解析

[5] The graphs of the lines y = x + 2 , y = 3 x + 4 , y = 5 x + 6 , y = 7 x + 8 , y = 9 x + 10 , y = 11 x + 12 are drawn. These six lines divide the plane into several regions. Compute the number of regions the plane is divided into. Proposed by: Arul Kolla Answer: 12 Solution: 4 2 y 0 − 2 − 4 − 3 − 2 − 1 0 1 2 x All lines are of the form y = xk + ( k + 1). Note that all lines pass through the point ( − 1 , 1), since 1 = ( − 1) k + ( k + 1) for all k . Thus all lines pass through a single point, ( − 1 , 1). The first line divides the plane into two parts, and each subsequent line divides two of the current regions into two more parts. Thus altogether the six lines divide the plane into 12 parts. 6