BuilderBoi

***EDIT*** 

I forgot to write the arithmetic series part correctly. The equation should be: 

$180(s-2) = s{(120 + (120 + 5(s-1)) \over 2}$

Solving, we find it can be 9, or 16. It can't be 16, because the largest angle is greater than 180, and then it would not be a convex polygon. 

Thus, the CORRECT answer is $\color{brown}\boxed{9}$

Thank You Geno for telling me!!!

Write as equation:

$180(s-2) = s({120 + (120+5s) \over 2})$

By solving, $\color{brown}\boxed{s = 12}$

It is the area bound by the 2 zeros, which are $s - 1 = 0$ and $s + 8 = 0$. 

Anything between these ranges work, thus the answer is $\color{brown}\boxed{-8 \le x \le 1}$

$y + 1 = 4x - 4$

$\color{brown}\boxed{y = 4x - 5}$

Only square numbers without any factor pairs (other than 1 and itself) work. These are the numbers: 4, 9, 25, 49, 121, and 169, making for a total of $\color{brown}\boxed{6}$ factors.

If x is positive, $x \over | x |$ will always equal 1. 

If x is 0, $0 \over 0$ will equal undefined

If x is negative, $x \over | x |$ will equal -1. 

Thus, there are $\color{brown}\boxed{3}$ distinct values. 

The slope of the line is -10/13. Multiply this by 10, and this means the line goes down -100/13 out of -130/13. This also means the line goes down 30/13 in the smaller square, meaning the area is: 

$9 - ({30 \over13} \times {3 \over 2}) = \color{brown}\boxed{72 \over 13}$

It can be 3,5,6, or 7. (It can't be 4, because that forms a 3-4-5 right triangle.) Thus, there are $\color{brown}\boxed{4}$ options. 

Using Chinese Remainder Theorem, the answer is $\color{brown}\boxed{1,067}$