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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.

# Absolute Value: Level 4 Challenges

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

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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