Question 124
A permutation of the integers $1$ to $2022$ is a sequence $\sigma = (\sigma_{1}, \ldots, \sigma_{2022})$ such that each element of the set $\{1, \ldots, 2022\}$ is equal to exactly one term $\sigma_{i}$. What is the smallest possible value that the sum
$$
\left\lfloor\frac{a_{1}}{1}\right\rfloor+\left\lfloor\frac{a_{2}}{2}\right\rfloor+\ldots+\left\lfloor\frac{a_{2022}}{2022}\right\rfloor
$$
can take when the integers $a_{1}, a_{2}, \ldots, a_{2022}$ form a permutation of the integers from $1$ to $2022$?
$$
\left\lfloor\frac{a_{1}}{1}\right\rfloor+\left\lfloor\frac{a_{2}}{2}\right\rfloor+\ldots+\left\lfloor\frac{a_{2022}}{2022}\right\rfloor
$$
can take when the integers $a_{1}, a_{2}, \ldots, a_{2022}$ form a permutation of the integers from $1$ to $2022$?