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# Absolute Value

Absolute value is a mathematician's way of judging numbers by their magnitude rather than their positive/negative value. It is the distance of the number from 0 on a number line.

\[\mathcal A=\displaystyle\sum_{i=1}^{2016}\left|x-a_i\right|\]

Let \(a_1,a_2,a_3,\ldots ,a_{2016}\) form an increasing arithmetic progression (AP) which consists of only positive terms. Let the minimum value of \(\mathcal A\) be \(2016^2\) for real \(x\). Then find the sum of all possible values of the common difference of AP.

**Notation**: \( | \cdot | \) denotes the absolute value function.

\[ \large x=|||\cdots |||2010-1|-2|-3|-\cdots -99|-100| \]

What is the value of \(x\)?

For real \(x\), find the minimum value of the expression below.

\[ |x-1| + |x-2| + \ldots + |x-100| \]

Define a function \(F\) as follows

\(F(w,x,y,z)=\dfrac { 1 }{ 8 } \left( S-T \right) \)

where

\( S=aw+bx+cy+dz-\left| w-x \right| -\left| aw+bx-cy-\left| w-x \right| \right| \)

\(T=\left| aw+bx+cy-dz-\left| w-x \right| -\left| aw+bx-cy-\left| w-x \right| \right| \right| \)

For certain positive integers \(a,b,c,d\) and for all \(w,x,y,z\), this function \(F\left(w,x,y,z\right)\) always returns the lowest of the values \(w,x,y,z\). \(\;\;\) For example

\(F(4,-3,2,1)=-3\).

Let \(a,b,c,d\) be integer digits of a \(4\) digit integer \(A = \overline{abcd} \). What is the value of \(A\)?

\[ \large |x - 1| + |2x-1| + |3x-1| + \cdots + |300x-1| \]

If the minimum value of the expression above is \( \dfrac AB\), and it occurs when \(x = \dfrac CD\), where \((A, B)\) and \((C, D)\) are each coprime pairs of positive integers, find \(A+B+C+D\).

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