HMMT 二月 2011 · 冲刺赛 · 第 4 题

HMMT February 2011 — Guts Round — Problem 4

[ 4 ] Let p be the answer to this question. If a point is chosen uniformly at random from the square bounded by x = 0, x = 1, y = 0, and y = 1, what is the probability that at least one of its coordinates is greater than p ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th 14 HARVARD-MIT MATHEMATICS TOURNAMENT, 12 FEBRUARY 2011 — GUTS ROUND

解析

[ 4 ] Let p be the answer to this question. If a point is chosen uniformly at random from the square bounded by x = 0, x = 1, y = 0, and y = 1, what is the probability that at least one of its coordinates is greater than p ? √ 5 − 1 Answer: 2 2 The probability that a randomly chosen point has both coordinates less than p is p , so the probability 2 that at least one of its coordinates is greater than p is 1 − p . Since p is the answer to this question, √ 5 − 1 2 we have 1 − p = p , and the only solution of p in the interval [0 , 1] is . 2