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max product of x_i when x_ ordered and sum x_1=100

Source: 2008 Swedish Mathematical Competition p6

April 27, 2021
algebrainequalitiesmaxProduct

Problem Statement

A sum decomposition of the number 100 is given by a positive integer nn and nn positive integers x1<x2<⋯<xnx_1<x_2<\cdots <x_n such that x1+x2+⋯+xn=100x_1 + x_2 + \cdots + x_n = 100. Determine the largest possible value of the product x1x2⋯xnx_1x_2\cdots x_n, and nn , as x1,x2,…,xnx_1, x_2,\dots, x_n vary among all sum decompositions of the number 100100.
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