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integers between $n^2$ and $n^2 + n + 3\sqrt{n}$

Source: 16-th Hungary-Israel Binational Mathematical Competition 2003

April 1, 2007

number theory unsolvednumber theory

Problem Statement

Let

n

be a positive integer. Show that there exist three distinct integers between

n^{2}

and

n^{2}+n+3\sqrt{n}

, such that one of them divides the product of the other two.