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 $a_{8} a_{17}>S+27, a_{11} a_{14}<S+60$. Indicate all possible values of $a_{1}$.
Answer: $-11; -10; -9; -7; -6; -5$.

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

$$
\left\{\begin{array} { l } 
{ ( a _ { 1 } + 7 d ) ( a _ { 1 } + 1 6 d ) > S + 2 7 , } \\
{ ( a _ { 1 } + 1 0 d ) ( a _ { 1 } + 1 3 d ) S+27 \\
a_{1}^{2}+23 a_{1} d+130 d^{2} 7 a _ { 1 } + 2 1 + 2 7 , } \\
{ a _ { 1 } ^ { 2 } + 2 3 a _ { 1 } + 1 3 0  0 , } \\
{ a _ { 1 } ^ { 2 } + 1 6 a _ { 1 } + 4 9 < 0 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
a_{1} \neq-8 \\
a_{1} \in(-8-\sqrt{15} ;-8+\sqrt{15})
\end{array}\right.\right.\right.
$$

Since $a_{1} \in \mathbb{Z}$, then $a_{1} \in\{-11; -10; -9; -7; -6; -5\}$.