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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.

Find the minimum value of \(\large \Bigg | \bigg | \Big | \big | x + 1 \big | + 2 \Big | + 3 \bigg | + 4 \Bigg | \) for real \(x\).

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

**Bonus**: Generalize this.

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

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

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

What is the value of \(x\)?

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

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.

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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 \(A\)?

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