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Smallest set that behaves like {1,2,...,n}

Source: Nordic MO 2012 Q3

April 21, 2013
modular arithmeticalgebra unsolvedalgebra

Problem Statement

Find the smallest positive integer nn, such that there exist nn integers x1,x2,…,xnx_1, x_2, \dots , x_n (not necessarily different), with 1≤xk≤n1\le x_k\le n, 1≤k≤n1\le k\le n, and such that x_1 + x_2 + \cdots + x_n =\frac{n(n + 1)}{2}, \text{ and }x_1x_2 \cdots x_n = n!, but {x1,x2,…,xn}≠{1,2,…,n}\{x_1, x_2, \dots , x_n\} \ne \{1, 2, \dots , n\}.
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