HMMT February

Algebra Difficulty 4.8 AIME Find the answer

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

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

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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Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.