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x_1x_2 ...x_n = x_1 + 2x_2 + 3x_3 +...+ nx_n

Source: 2016 Romanian NMO grade VIII P4

September 3, 2024
number theory

Problem Statement

For n∈N∗n \in N^* we will say that the non-negative integers x1,x2,...,xnx_1, x_2, ... , x_n have property (P)(P) if x1x2...xn=x1+2x2+3x3+...+nxn.x_1x_2 ...x_n = x_1 + 2x_2 + 3x_3 + ...+ nx_n.
a) Show that for every n∈N∗n \in N^* there exists nn positive integers with property (P)(P).
b) Find all integers n≥2n \ge 2 so that there exists nn positive integers x1,x2,...,xnx_1, x_2, ... , x_n with x1<x2<x3<...<xnx_1< x_2<x_3< ... <x_n, having property (P)(P).
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