Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.6 AIME Prove it Saudi Arabia

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

Solution

Suppose that we have
$\frac{1 + \sqrt{8n + 1}}{2} < x < \frac{1 + \sqrt{8n + 9}}{2}$
for some positive integer $x$ . Then
$\sqrt{8n + 1} < 2x - 1 < \sqrt{8n + 9}$
hence $8n + 1 < 4x^{2} - 4x + 1 < 8n + 9$ . It follows $2n < x^{2} - x < 2n + 2$ , that is $x^{2} - x = 2n + 1$ , not possible since $x^{2} - x$ is an even integer.

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