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Solve [x/2]+[x/3] = x-2022

Source: Cyprus 2022 Junior TST-2 Problem 1

February 21, 2022
algebrafloor functionequation

Problem Statement

Determine all real numbers x∈Rx\in\mathbb{R} for which ⌊x2⌋+⌊x3⌋=x−2022. \left\lfloor \frac{x}{2} \right\rfloor + \left\lfloor \frac{x}{3} \right\rfloor=x-2022. The notation ⌊z⌋\lfloor z \rfloor, for z∈Rz\in\mathbb{R}, denotes the largest integer which is less than or equal to zz. For example: \lfloor 3.98 \rfloor =3   \text{and}   \lfloor 0.14 \rfloor =0.
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