Question 143
Denote by $\lfloor x\rfloor$ the greatest integer $\leq x$.
Let $\lambda \geq 1$ be a real number and $n$ a positive integer such that $\left\lfloor\lambda^{n+1}\right\rfloor,\left\lfloor\lambda^{n+2}\right\rfloor, \ldots,\left\lfloor\lambda^{4 n}\right\rfloor$ are all perfect squares. Prove that $\lfloor\lambda\rfloor$ is a perfect square.
Let $\lambda \geq 1$ be a real number and $n$ a positive integer such that $\left\lfloor\lambda^{n+1}\right\rfloor,\left\lfloor\lambda^{n+2}\right\rfloor, \ldots,\left\lfloor\lambda^{4 n}\right\rfloor$ are all perfect squares. Prove that $\lfloor\lambda\rfloor$ is a perfect square.