HMMT 二月 2017 · 冲刺赛 · 第 12 题

HMMT February 2017 — Guts Round — Problem 12

[ 7 ] In a certain college containing 1000 students, students may choose to major in exactly one of math, computer science, finance, or English. The diversity ratio d ( s ) of a student s is the defined as number of students in a different major from s divided by the number of students in the same major as s (including s ). The diversity D of the college is the sum of all the diversity ratios d ( s ). Determine all possible values of D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . February 2017, February 18, 2017 — GUTS ROUND Organization Team Team ID#

解析

[ 7 ] In a certain college containing 1000 students, students may choose to major in exactly one of math, computer science, finance, or English. The diversity ratio d ( s ) of a student s is the defined as number of students in a different major from s divided by the number of students in the same major as s (including s ). The diversity D of the college is the sum of all the diversity ratios d ( s ). Determine all possible values of D . Proposed by: Evan Chen Answer: { 0 , 1000 , 2000 , 3000 } It is easy to check that if n majors are present, the diversity is 1000( n − 1). Therefore, taking n = 1 , 2 , 3 , 4 gives us all possible answers.