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Algebra Difficulty 6.6 National Olympiad Prove it Ukraine

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.

Solution

$\frac{2010 - (2009 - 2010 - 2009 - 2010 - 2009 - \dots - 2010 - 2009)}{2010 \text{ numbers}} = 4035077.$

Solution. Consider an arbitrary arrangement of the brackets. The first numbers $2010$ and $2009$ always have signs '+' and '-', respectively, that's why the sum reaches the maximum value when other terms will have '+' signs. We can achieve this as it was made in the answer. Calculating maximum value:
$\begin{align*} & \underbrace{2010 - (2009 - 2010 - 2009 - 2010 - 2009 - \dots - 2010 - 2009)}_{2010 \text{ numbers}} = \\ &= 2010 - 2009 + \underbrace{2010 + 2009 + 2010 + 2009 + \dots + 2010 + 2009}_{2008 \text{ numbers}} = \\ &= 1 + (2010 + 2009) \cdot 1004 = 4035077. \end{align*}$

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