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 \(\color{brown}\boxed{1,067}\)