Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.9 AIME Prove it United States

Problem:
Find all real numbers $x$ such that
$x^{2} + \left\lfloor \frac{x}{2} \right\rfloor + \left\lfloor \frac{x}{3} \right\rfloor = 10$

Solution

Solution:
Evidently $x^{2}$ must be an integer. Well, there aren't that many things to check, are there? Among positive $x$ , $\sqrt{8}$ is too small and $\sqrt{9}$ is too big; among negative $x$ , $-\sqrt{15}$ is too small and $-\sqrt{13}$ is too big.

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