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, 2017

Algebra Difficulty 5.4 AIME, harder Prove it United States

Problem:

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?

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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