$\mathrm{Zij} n \geq 1$. Find all permutations $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ of $(1,2, \ldots, n)$ for which

$$
\frac{a_{k}^{2}}{a_{k+1}} \leq k+2 \quad \text { for } k=1,2, \ldots, n-1
$$
Suppose for some $i \leq n-1$ it holds that $a_{i}>i$. It follows that $\frac{a_{i}^{2}}{a_{i+1}} \leq i+2$, so

$$
a_{i+1} \geq \frac{a_{i}^{2}}{i+2} \geq \frac{(i+1)^{2}}{i+2}=\frac{i^{2}+2 i+1}{i+2}>i.
$$

If $a_{i} \geq i+2$, then in the same way $a_{i+1}>i+1$. If $a_{i}=i+1$, then $i+1$ is no longer available as a value for $a_{i+1}$, so $a_{i+1}>i+1$ also holds.
Thus, if $a_{i}>i$, then also $a_{i+1}>i+1$. Since $a_{n} \leq n$, there can be no $i$ such that $a_{i}>i$. It follows that $a_{i}=i$ for all $i$.