Estonian Mathematical Olympiad

We will show that a value greater than $8$ is impossible. For this, let us assume the opposite. In this case, we can simplify the expression by removing the $0$ and the operation to the left of it. The remaining operation consists of the numbers $-2,-1,1,2$ and three operations and has a value of at least $9$. We will consider two cases.

If the rightmost number of the simplified expression is $2$ or $-2$, then the absolute value of the subexpression to the left of it needs to be at least $5$. Then, if the second number from the right is also $2$ or $-2$, then the absolute value of the subexpression to the left of it needs to be at least $3$, which is impossible using just the numbers $-1$ and $1$. Similarly, if the second number from the right is $-1$ or $1$, then the absolute value of the subexpression to the left of it needs to be at least $4$, which is impossible using two numbers with absolute values $1$ and $2$.

If the rightmost number of the simplified expression is $1$ or $-1$, then the absolute value of the subexpression to the left of it needs to be at least $8$. Then, if the second number from the right is also $1$ or $-1$, then the absolute value of the subexpression to the left of it needs to be at least $7$, which is impossible using just the numbers $-2$ and $2$. Similarly, if the second number from the right is $-2$ or $2$, then the absolute value of the subexpression to the left of it needs to be at least $4$, which is impossible using two numbers with absolute values $1$ and $2$.

We obtained a contradiction in all cases, which means that values greater than $8$ are impossible.

On the other hand, if $a=-1$, $b=1$, $c=-2$, $d=2$, $e=0$, then $(((a-b)\cdot c)\cdot d)+e=8$.