GR0001

This answer is going to assume that the short side of the rectange is equal to the radius of the large circle.

Our goal is to find the short side of the rectangle. The short side must be between 6 and 12(radius and diameter of small circle). Let's start by finding some factor pairs for 243.

Pair 1: 1*243 (doesn't work)

Pair 2: 3*81(doesn't work)

Pair 3: 9*27(works)

As those are the only three pairs, pair 3 will work.

Now that we know the short side is 9, we can see that the radius of the large circle is 9

Found the answer after working on the problem for a bit:

Pick 5 states out of the 50 possible, 50 choose 5 (2118760)

Choose 1 out of 2 competitors out of each state. 

2⁵ because there are 5 states and 2 choices for each.

2118760*32=67800320

Found it!

Pick 5 states out of the 50 possible, 50 choose 5 (2118760)

Choose 1 out of 2 competitors out of each state. 

2⁵ because there are 5 states and 2 choices for each.

2118760*32=67800320

That's okay, I'll keep working to see how to arrive at 67,800,320. Thanks anyways though!

hihihi's solution is good, but this one may help you solve the problem quicker and easier

(x+y)² = x²+2xy+y²

(x-y)²=x²-2xy+y²

As such, when you add them together, you get 2x²+2y²

When you plug in x and y, this equals to 2(2006)+2(2007)=4012+4014=8026

Wait, but according to the official national MATHCOUNTS solutions, the answer is 67,800,320 ways, was there something wrong in your solution?

(x+y)² = x²+2xy+y²

(x-y)²=x²-2xy+y²

As such, when you add them together, you get 2x²+2y²

When you plug in x and y, this equals to 2(2006)+2(2007)=4012+4014=8026

What I like about this question is that you can set an amount for the stock price to easily solve it. Let's set the stock price to 100.

First day: lose 10%

100-(10%)100=100-10=90

Second day: lose 20%

90-(20%)90=90-18=72

Third day: lose 30%

72-(30%)72=72-21.6=50.4

In total, you lost 49.6$ over the whole period. As 49.6 is 49.6% of 100, you lost a total of 49.6%