Question 124
Consider an arbitrary arrangement of the brackets. Numbers $2010$ and $2009$ always have signs '+' and '-', respectively, that's why the sum reaches the maximum value when the other terms will have '+' sign.
In the expression
$$
\frac{2010 - 2009 - 2010 - 2009 - 2010 - 2009 - \dots - 2010 - 2009}{2010 \text{ numbers}}
$$
there are somehow placed brackets and the value is calculated. Find the maximum value that can be reached. Justify your answer.
*Notice.* Left bracket can be placed only before a number and right - only after. For example, expressions $-2010(-2009 - 2010)$ and $-(2010 - 2009-)2010$ are incorrect.
In the expression
$$
\frac{2010 - 2009 - 2010 - 2009 - 2010 - 2009 - \dots - 2010 - 2009}{2010 \text{ numbers}}
$$
there are somehow placed brackets and the value is calculated. Find the maximum value that can be reached. Justify your answer.
*Notice.* Left bracket can be placed only before a number and right - only after. For example, expressions $-2010(-2009 - 2010)$ and $-(2010 - 2009-)2010$ are incorrect.