I'm back!

After much moping about the house, I'm finally back in my math groove! Except....now my LaTeX image generator isn't working anymore!!! T_T. Luckily this one doesn't require much latex, so I'll just type it out:
A function f is defined by f(z) = (4+i)z^2 + az + y, where a and y are complex numbers. Given that f(1) and f(i) are both real, what is the minimum value of |a| + |y|?

They also mentioned something about how i^2 = -1. If you didn't know that then I would recommend you close out the browser and go open a middle school math textbook.

So how should we start? I'll tell you something I've noticed. The AMC problems are meant to look intimidating. They really aren't that bad. So we are given that certain values have no imaginary component. This is a clue to split a and y into real and imaginary components:

a := a + bi
y := y + zi

I used := because I mean that we're completely redefining them now.

So lets look at f(1) and f(i):

f(1) = 4 + i + a + bi + y + zi

From this we can conclude that b + z = -1, as we need all the i terms to go away

f(i) = -4 - i + ai - b + yi - z

Again, we conclude that a + y = 1

This is where most people would get stuck. How can we possible find the minimum value of |a| + |y| given this information? Here, I turned to a geometric interpretation using vectors. In AP Physics we've been doing vector component stuff. If you interpret a and y as the x components and b and y and z as the y components, then we get a picture like this:




I thought that would be too big but now it looks too small.

Well anyway, the point is that if we look at the complex numbers as vectors, their x and y components both add up to 1. Well, technically one of them adds up to -1, but it doesn't matter. So the absolute value of each will be the length of each vector added together. However, since we know how their components add up, we know that if we add the vectors together using that silly head to tail method, they go to the point (-1,1). What's the shortest path from 1 point to another? A straight line. The distance from the origin to (-1,1) is the square root of 2. Problem solved!