Question 5
Let $n$ be a positive integer. For a real $x \ge 1$, assume that $\lfloor x^{n+1} \rfloor, \lfloor x^{n+2} \rfloor, \dots, \lfloor x^{4n} \rfloor$ are all squares of positive integers. Prove that $\lfloor x \rfloor$ is also the square of a positive integer.
Here $\lfloor z \rfloor$ is the greatest integer smaller than or equal to $z$.
Here $\lfloor z \rfloor$ is the greatest integer smaller than or equal to $z$.