I just took a quiz and am dumbfounded by my lack of insight. Consider what kind of idiot I'd have to be to do the following:
Point A = (8,-15) and point B = (-8,15). P is the locus of points (x,y) such that AP • BP = 0. Describe the elements in P.
Immediately I graphed A and B and then observed that their dot product was 0, which meant that vectors AP and BP are perpendicular. Now, out of some stupidity, I immediately drew the vertical and horizontal lines through A and B and concluded that the two points of intersection are the only elements of the locus. Clearly, though, the locus is a circle, because if we inscribe a right angle into a circle the hypotenuse is the diameter. Since the intersection of A and B is a right angle, the diameter is AB. The radius is the distance between A and B divided by 2, which is 17. The locus is thus the circle x^2 + y^2 = 289. A simple, easy problem with an intuitive solution.
Why did I do this incorrectly? How can I think better in test situations and at home (similarly, I find myself spending 2-3 hours on math homework daily while the A+ student in my class spends 45 minutes maximum)? Indeed, in regular math I got an 100 on every single assessment without exception. But this is plug and chug pseudo-math. Honors math is the real thing - learning concepts and using them to see connections and draw new things based on old knowledge. How effectively one does this determines the grade in honors math.
The bottom line is - how do I get amazing at real math? How do I think and realize things during a quiz or test? How do I come up with ideas about new problems related to old ideas? Really - how do I become mathematically intelligent?
If you are interested in mathematics, as you seem to be, I suggest you start solving problems in your free time as a hobby. You will get better and learn many fundamental strategies that do not come from studying. A good book on problem solving is "arts and crafts of problem solving" by Paul Zeitz.
A problem you can work on below:
n points are placed on a circle, what is the maximum number of regions you can divide the circle into by connecting these points.
(hint : not $2^{n-1}$)