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Show that for all natural number n

Source: Iranian 3rd round Number Theory exam P1

September 22, 2014
number theory proposednumber theory

Problem Statement

Show that for every natural number nn there are nn natural numbers x1<x2<...<xn x_1 < x_2 < ... < x_n such that
1x1+1x2+...+1xn−1x1x2...xn∈N∪0\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}-\frac{1}{x_1x_2...x_n}\in \mathbb{N}\cup {0}
(15 points )
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