1. [5 points] $S$ - the sum of the first 7 terms of an increasing arithmetic progression $a_{1}, a_{2}, a_{3}, \ldots$, consisting of integers. It is known that $a_{7} a_{12}>S+20, a_{9} a_{10}<S+44$. Indicate all possible values of $a_{1}$.
Answer: $-9; -8; -7; -6; -4; -3; -2; -1$.

Solution. Let the common difference of the progression be $d$. The inequalities given in the condition can be transformed as follows:

$$
\left\{\begin{array} { l } 
{ ( a _ { 1 } + 6 d ) ( a _ { 1 } + 11 d ) > S + 20 , } \\
{ ( a _ { 1 } + 8 d ) ( a _ { 1 } + 9 d ) S+20 \\
a_{1}^{2}+17 a_{1} d+72 d^{2} 7 a _ { 1 } + 21 + 20 , } \\
{ a _ { 1 } ^ { 2 } + 17 a _ { 1 } + 72  0 , } \\
{ a _ { 1 } ^ { 2 } + 10 a _ { 1 } + 7 < 0 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
a_{1} \neq-5 \\
a_{1} \in(-5-\sqrt{18} ;-5+\sqrt{18})
\end{array}\right.\right.\right.
$$

Since $a_{1} \in \mathbb{Z}$, then $a_{1} \in\{-9; -8; -7; -6; -4; -3; -2; -1\}$.