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HMMT 十一月 2016 · 冲刺赛 · 第 23 题

HMMT November 2016 — Guts Round — Problem 23

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 12 ] Let d be a randomly chosen divisor of 2016. Find the expected value ofd 2
    d + 20162
解析

英文解析

  1. [ 12 ] Let d be a randomly chosen divisor of 2016. Find the expected value ofd 2
    d + 20162
    Proposed by: Henrik Boecken
    Answer:1
    Let ab = 2016. Then 2
    2016
    2 2 2 22
    ( )
    a b a a 2016 + = + = + = 1 a
    2016
    2 2 2 2 2
    a + 2016 b + 2016 a + 2016 a + 2016 a + 20162
    ( ) + 2016
    1 a
    2016
    Thus, every divisor d pairs up with to get 1, so our desired expected value is .
    2 d