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Filling Boxes

Source: The 1st India-Iran Friendly Competition Problem 5

June 13, 2024
combinatoricsalgorithms

Problem Statement

Let n≥kn \geq k be positive integers and let a1,…,ana_1, \dots, a_n be a non-increasing list of positive real numbers. Prove that there exists kk sets B1,…,BkB_1, \dots, B_k which partition the set {1,2,…,n}\{1, 2, \dots, n\} such that min⁡1≤j≤k(∑i∈Bjai)≥min⁡1≤j≤k(12k+1−2j⋅∑i=jnai).\min_{1 \le j \le k} \left(\sum_{i \in B_j} a_i \right) \geq \min_{1 \le j \le k} \left(\frac{1}{2k+1-2j} \cdot \sum^n_{i=j} a_i\right).
Proposed by Navid Safaei
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