IMC 2025 · Problem 4

Let $a$ be an even positive integer. Find all real numbers $x$ such that

$$\left\lfloor \sqrt[a]{b^a + x} \cdot b^{a-1} \right\rfloor = b^a + \lfloor x/a \rfloor \tag{1}$$

holds for every positive integer $b$.

(Here $\lfloor x \rfloor$ denotes the largest integer that is no greater than $x$.)