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Maximizing a sum !

Source: Romania TST 2015 Day 1 Problem 5

April 9, 2015
maximizationalgebrainequalitiesSum

Problem Statement

Given an integer N≥4N \geq 4, determine the largest value the sum ∑i=1⌊k2⌋+1(⌊ni2⌋+1)\sum_{i=1}^{\left \lfloor{\frac{k}{2}}\right \rfloor+1}\left( \left \lfloor{\frac{n_i}{2}}\right \rfloor+1\right) may achieve, where k,n1,…,nkk, n_1, \ldots, n_k run through the integers subject to k≥3k \geq 3, n1≥…≥nk≥1n_1 \geq \ldots\geq n_k\geq 1 and n1+…+nk=Nn_1 + \ldots + n_k = N.
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