2014 IMO Shortlist N4

Let $n>1$ be a given integer. Prove that infinitely many terms of the sequence $\left(a_{k}\right)_{k \geq 1}$, defined by $$a_{k}=\left\lfloor\frac{n^{k}}{k}\right\rfloor$$ are odd. (For a real number $x,\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) (Hong Kong)

令 $n>1$ 为给定整数。证明由 $$a_{k}=\left\lfloor\frac{n^{k}}{k}\right\rfloor$$ 定义的序列 $\left(a_{k}\right)_{k \geq 1}$ 的无穷多个项是奇数。 (对于实数$x，\lfloor x\rfloor$表示不超过$x$的最大整数。)(香港)