HMMT 十一月 2015 · GEN 赛 · 第 2 题

HMMT November 2015 — GEN Round — Problem 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.

解析

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 Katz π − 2 Answer: 4 2 2 We require a + b > 1 and a + b < 1. Geometrically, this is the area enclosed in the quarter-circle centered at the origin with radius 1, not including the area enclosed by a + b < 1 (an isosceles right π − 2 triangle with side length 1). As a result, our desired probability is . 4