HMMT 二月 2015 · 几何 · 第 1 题

HMMT February 2015 — Geometry — Problem 1

Let R be the rectangle in the Cartesian plane with vertices at (0 , 0) , (2 , 0) , (2 , 1) , and (0 , 1). R can be divided into two unit squares, as shown. Pro selects a point P uniformly at random in the interior of R . Find the probability that the line 1 through P with slope will pass through both unit squares. 2

解析

Let R be the rectangle in the Cartesian plane with vertices at (0 , 0) , (2 , 0) , (2 , 1) , and (0 , 1). R can be divided into two unit squares, as shown. Pro selects a point P uniformly at random in the interior of R . Find the probability that the line 1 through P with slope will pass through both unit squares. 2 3 Answer: 4 Precisely the middle two (of four) regions satisfiy the problem conditions, and it’s easy to compute the 3 (fraction of) areas as . 4