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HMMT 二月 2015 · 几何 · 第 1 题

HMMT February 2015 — Geometry — Problem 1

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

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

英文解析

  1. Let R be the rectangle in the Cartesian plane with vertices at (0 , 0) , (2 , 0) , (2 , 1) , and (0 , 1). R can bedivided into two unit squares, as shown.
    Pro selects a point P uniformly at random in the interior of R . Find the probability that the linethrough P with slope will pass through both unit squares.1
    Answer:32
    Precisely the middle two (of four) regions satisfiy the problem conditions, and it’s easy to compute the 4
    (fraction of) areas as .3 4