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Problem 1321

AIME late

Algebra Difficulty 5.4 Prove it February · United States · 2017

Mrs. Toad has a class of 2017 students, with unhappiness levels $1,2, \ldots, 2017$ 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?

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Solution:

One can show that the optimal configuration is $\{1\}, \{2\}, \ldots, \{14\}, \{15, \ldots, 2017\}$ . This would give us an answer of $1+2+\cdots+14+\frac{15+2017}{2}=105+1016=1121$ .

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