HMMT 二月 2017 · 冲刺赛 · 第 14 题

HMMT February 2017 — Guts Round — Problem 14

[ 9 ] Mrs. Toad has a class of 2017 students, with unhappiness levels 1 , 2 , . . . , 2017 respectively. Today in class, there is a group project and Mrs. Toad wants to split the class in exactly 15 groups. The unhappiness level of a group is the average unhappiness of its members, and the unhappiness of the class is the sum of the unhappiness of all 15 groups. What’s the minimum unhappiness of the class Mrs. Toad can achieve by splitting the class into 15 groups?

解析

[ 9 ] Mrs. Toad has a class of 2017 students, with unhappiness levels 1 , 2 , . . . , 2017 respectively. Today in class, there is a group project and Mrs. Toad wants to split the class in exactly 15 groups. The unhappiness level of a group is the average unhappiness of its members, and the unhappiness of the class is the sum of the unhappiness of all 15 groups. What’s the minimum unhappiness of the class Mrs. Toad can achieve by splitting the class into 15 groups? Proposed by: Yang Liu One can show that the optimal configuration is { 1 } , { 2 } , . . . , { 14 } , { 15 , . . . , 2017 } . This would give us 15+2017 an answer of 1 + 2 + · · · + 14 + = 105 + 1016 = 1121 . 2