HMMT 二月 2006 · 冲刺赛 · 第 42 题

HMMT February 2006 — Guts Round — Problem 42

[18] Suppose hypothetically that a certain, very corrupt political entity in a universe holds an election with two candidates, say A and B . A total of 5,825,043 votes are cast, but, in a sudden rainstorm, all the ballots get soaked. Undaunted, the election officials decide to guess what the ballots say. Each ballot has a 51% chance of being deemed a vote for A , and a 49% − X chance of being deemed a vote for B . The probability that B will win is 10 . What is X rounded to the nearest 10? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th IX HARVARD-MIT MATHEMATICS TOURNAMENT, 25 FEBRUARY 2006 — GUTS ROUND

解析

Suppose hypothetically that a certain, very corrupt political entity in a universe holds an election with two candidates, say A and B . A total of 5,825,043 votes are cast, but, in a sudden rainstorm, all the ballots get soaked. Undaunted, the election officials decide to guess what the ballots say. Each ballot has a 51% chance of being deemed a vote for A , and a 49% chance of being deemed a vote for B . The probability that B − X will win is 10 . What is X rounded to the nearest 10? Answer: 510 Solution: Let N = 2912521, so that the number of ballots cast is 2 N + 1. Let P be the probability that B wins, and let α = 51% and β = 49% and γ = β/α < 1. We have ( ) ( ) N N ∑ ∑ 2 N + 1 2 N + 1 − X N − i N +1+ i N N +1 i 10 = P = α β = α β γ N − i N − i i =0 i =0 (think of 2 i + 1 as representing B ’s margin of victory). Now ( ) ( ) N 2 N +1 ∑ 2 2 N + 1 2 N + 1 i 2 N +1 < < γ < 2 , 2 N + 1 N N − i i =0 15 So − X = log P = N log α +( N +1) log β +(2 N +1) log 2 − = N log(2 α )+( N +1) log(2 β ) − , where 0 < < log(2 N + 1) < 7. With a calculator, we find that − X ≈ 25048 . 2 − 25554 . 2 − = − 506 . 0 − , so X ≈ 510.