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an interval does not contain any integer

Source: 2011 Saudi Arabia Pre-TST February 2.1

January 1, 2022
number theoryInteger

Problem Statement

Let nn be a positive integer. Prove that the interval In=(1+8n+12,1+8n+92)I_n= \left( \frac{1+\sqrt{8n+1}}{2}, \frac{1+\sqrt{8n+9}}{2}\right) does not contain any integer.
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