HMMT 二月 2002 · 几何 · 第 6 题

HMMT February 2002 — Geometry — Problem 6

If we pick (uniformly) a random square of area 1 with sides parallel to the x − and y − axes that lies entirely within the 5-by-5 square bounded by the lines x = 0 , x = 5 , y = 0 , y = 5 (the corners of the square need not have integer coordinates), what is the probability that the point ( x, y ) = (4 . 5 , 0 . 5) lies within the square of area 1?

解析

If we pick (uniformly) a random square of area 1 with sides parallel to the x − and y − axes that lies entirely within the 5-by-5 square bounded by the lines x = 0 , x = 5 , y = 0 , y = 5 (the corners of the square need not have integer coordinates), what is the probability that the point ( x, y ) = (4 . 5 , 0 . 5) lies within the square of area 1? Solution: The upper-left corner of the unit square is picked uniformly from the square 0 ≤ x ≤ 4; 1 ≤ y ≤ 5, and for it to contain the desired point it must lie in the square 1 1