1.  This is the same as asking how many points are on the line, and we know
that there are 97 points on the given line, mod 97.

2.  By trial and error, we get x = 6.

3.  By trial and error, we get x = 2, x = 5 (more than 1 solution since mod 6
is not a field)

4a. y = 3, y = 3x+3

4b.  (3,0) (3,1) (3,2) (3,3) (3,4)

5. Here there are 8 lines,  we have x=2, y=3 y=x+1, y = 2x +6, y = 3x +4,
y = 4x + 2, y = 5x, y = 6x +5.
Basically list the lines that we already know, and then we know to use the
basic form of y=mx+b.  Using the point (2,3), you can plug in values for m, and
solve for b. 

6. 36*c = 24*6 => c = 4 

7a.  This field has 16 elements.

b.  t^3 + t

c.  16^2 +16 lines, 16 lines on a point.

8.  37^2 + 37 lines.

9.  Plug in 0,1,2,3,4 for x, and the polynomial is never equal to 0.

10. 5*C/25 = C/5 (contests * player per contests / contestants)

(5 choose 2) * C / (25 choose 2) = C/30