HMMT 二月 2018 · 冲刺赛 · 第 15 题

HMMT February 2018 — Guts Round — Problem 15

[ 9 ] Michael picks a random subset of the complex numbers { 1 , ω, ω , . . . , ω } where ω is a primitive th 2018 root of unity and all subsets are equally likely to be chosen. If the sum of the elements in his 2 subset is S , what is the expected value of | S | ? (The sum of the elements of the empty set is 0.)

解析

[ 9 ] Michael picks a random subset of the complex numbers { 1 , ω, ω , . . . , ω } where ω is a primitive th 2018 root of unity and all subsets are equally likely to be chosen. If the sum of the elements in his 2 subset is S , what is the expected value of | S | ? (The sum of the elements of the empty set is 0.) Proposed by: Nikhil Reddy 1009 Answer: 2 Consider a and − a of the set of complex numbers. If x is the sum of some subset of the other complex numbers, then expected magnitude squared of the sum including a and − a is ( x + a )( x + a ) + xx + xx + ( x − a )( x − a ) 4 aa xx + 2 1 xx + 2 1 By repeating this process on the remaining 2016 elements of the set, we can obtain a factor of every 2 time. In total, the answer is 1009 2