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HMMT 二月 2011 · 冲刺赛 · 第 4 题

HMMT February 2011 — Guts Round — Problem 4

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

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

英文原题

[ 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 ?
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14 th HARVARD-MIT MATHEMATICS TOURNAMENT, 12 FEBRUARY 2011 — GUTS ROUND

解析

英文解析

  1. [ 4 ] Let p be the answer to this question. If a point is chosen uniformly at random from the squarebounded by x = 0, x = 1, y = 0, and y = 1, what is the probability that at least one of its coordinatesis greater than p ?
    √
    5 − 1
    Answer:
    The probability that a randomly chosen point has both coordinates less than p is p , so the probability 22
    that at least one of its coordinates is greater than p is 1 − p . Since p is the answer to this question,2
    √
    5 − 1
    we have 1 − p = p , and the only solution of p in the interval [0 , 1] is .2 2