SE-SKOLORNAS 2008 Problem 6

A sum decomposition of the number 100 is given by a positive integer $n$ and $n$ positive integers $x_1<x_2<\cdots <x_n$ such that $x_1 + x_2 + \cdots + x_n = 100$ . Determine the largest possible value of the product $x_1x_2\cdots x_n$ , and $n$ , as $x_1, x_2,\dots, x_n$ vary among all sum decompositions of the number $100$ .