HMMT 十一月 2015 · GEN 赛 · 第 2 题

英文解析

2. Let a and b be real numbers randomly (and independently) chosen from the range [0 , 1]. Find the probability that a, b and 1 form the side lengths of an obtuse triangle.
   Proposed by: Alexander Katzπ − 2
   Answer:
   2 24
   We require a + b > 1 and a + b < 1. Geometrically, this is the area enclosed in the quarter-circlecentered at the origin with radius 1, not including the area enclosed by a + b < 1 (an isosceles rightπ − 2
   triangle with side length 1). As a result, our desired probability is . 4