2021 APMO 第 1 题

Prove that for each real number $r>2$, there are exactly two or three positive real numbers $x$ satisfying the equation $x^{2}=r\lfloor x\rfloor$.

Note: $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$

证明对于每个实数$r>2$，恰好有两个或三个正实数$x$满足方程$x^{2}=r\lfloor x\rfloor$。

Note: $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$