You know, It’s funny. When I sit down to write one of these, I usually have a rough sketch in my mind of how the question’s going to go. But half the time, the problem completely evolves into something else by the time I’m done with it. I’m not sure why that is. It could just be that I’m capricious, but it’s probably because math comes into much sharper focus once it’s written down. Which I only mention to gently encourage you to write everything down when you take tests.
Anyway, you know the drill by now. First correct response in the comments wins a copy of my Math Guide. Full contest rules here.
Note: If you’re new to the site, your comment might not appear right away. Don’t panic—I receive comments in my email in the order they’re submitted. If you’re the first to get it right, you’ll win even if your comment doesn’t immediately appear.
In the figure above, A is the vertex of f(x), B and C are the x-intercepts of f(x), F is the origin, D is the midpoint of AF, and E is the midpoint of DF. If f(x) = px2 + 8 and the area of concave quadrilateral BDCE is equal to 8, what is p?
UPDATE: Peter got it first. Solution below the cut.
The first thing to focus on here is the equation, f(x) = px2 + 8, which tells you a very important thing: the y-intercept of f(x) is 8. This is one of the few things you need to keep in mind about parabolas—that the constant in a parabola equation tells you its y-intercept. This makes sense, of course, because the y-intercept is when x = 0, and f(0) = p(0)2 + 8 = 8. So yeah. Point A is (0, 8).
We know that D and E are midpoints, so we now know a few distances. Let’s throw them on the figure: