Let $f_0(x)=x+|x-100|-|x+100|$, and for $n\geq 1$, let $f_n(x)=|f_{n-1}(x)|-1$. For how many values of $x$ is $f_{100}(x)=0$?

Suppose that $\left|x+y\right|+\left|x-y\right|=2$. What is the maximum possible value of $x^2-6x+y^2$?

If $x<0$, then which of the following must be positive?

What is the minimum value of $\left|x-1\right| + \left|2x-1\right| + \left|3x-1\right| + \cdots + \left|119x - 1 \right|$?

What is the area of the region defined by the inequality $|3x-18|+|2y+7|\le3$?

What is the product of all the roots of the equation \[\sqrt{5 | x | + 8} = \sqrt{x^2 - 16}.\]

Let $a$, $b$, $c$, and $d$ be real numbers with $|a-b|=2$, $|b-c|=3$, and $|c-d|=4$. What is the sum of all possible values of $|a-d|$?

What is the area of region bounded by the graphs of $y=|x+2| -|x-2|$ and $y=|x+1|-|x-3|$?

Two different positive numbers $a$ and $b$ each differ from their reciprocals by $1$. What is $a+b$?

Let $x=-2016$. What is the value of $\bigg|$ $|x|-x|-|x|$ $\bigg|$ $-x$?

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