10.3. $\left(a_{n}\right)$ is an arithmetic progression with a common difference of 1. It is known that $\mathrm{S}_{2022}$ is the smallest among all $S_{n}$ (less than the sum of the first $n$ terms for any other value of n). What values can the first term of the progression take?
Answer: $a_{1}$ belongs to the interval ( $-2022 ; -2021$ ).

Solution. Since the difference of the progression is positive, the progression is increasing. Therefore, the described situation is possible if and only if the members of the progression from the first to the 2022nd are negative, and starting from the 2023rd they are positive. Thus, $\mathrm{S}_{2022}$ will be the smallest if and only if $a_{2022} < 0$ and $a_{2023} > 0$. From this, we obtain the system of inequalities

$\left\{\begin{array}{l}a_{1} + 2021d < 0 \\ a_{1} + 2022d > 0\end{array} \Leftrightarrow -2022 < a_{1} < -2021\right.$.