Absolute Value

Absolute Value: Level 4 Challenges


A=i=12016xai\mathcal A=\displaystyle\sum_{i=1}^{2016}\left|x-a_i\right|

Let a1,a2,a3,,a2016a_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 A\mathcal A be 201622016^2 for real xx. Then find the sum of all possible values of the common difference of AP.

Notation: | \cdot | denotes the absolute value function.

x=201012399100 \large x=|||\cdots |||2010-1|-2|-3|-\cdots -99|-100|

What is the value of xx?

For real xx, find the minimum value of the expression below.

x1+x2++x100 |x-1| + |x-2| + \ldots + |x-100|

Define a function FF as follows

F(w,x,y,z)=18(ST)F(w,x,y,z)=\dfrac { 1 }{ 8 } \left( S-T \right)


S=aw+bx+cy+dzwxaw+bxcywx S=aw+bx+cy+dz-\left| w-x \right| -\left| aw+bx-cy-\left| w-x \right| \right|
T=aw+bx+cydzwxaw+bxcywxT=\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,da,b,c,d and for all w,x,y,zw,x,y,z, this function F(w,x,y,z)F\left(w,x,y,z\right) always returns the lowest of the values w,x,y,zw,x,y,z.     \;\; For example


Let a,b,c,da,b,c,d be integer digits of a 44 digit integer A=abcdA = \overline{abcd} . What is the value of AA?

x1+2x1+3x1++300x1 \large |x - 1| + |2x-1| + |3x-1| + \cdots + |300x-1|

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


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