IMC 2025 Problem 4

Let $a$ be an even positive integer. Find all real numbers $x$ such that $$\left\lfloor\sqrt[a]{b^a+x} \cdot b^{a-1}\right\rfloor=b^a+\lfloor x / a\rfloor$$ holds for every positive integer $b$ . (Here $\lfloor x\rfloor$ denotes the largest integer that is no greater than $x$ .)