Imnotamaster

Let's say smaller leg is X

The longer leg is x+8

Hypo is x+14 

a^2 + b^2 = c

so x^2 + (x+8)^2 = (x+14)^2. 
Expand
x^2 + x^2 + 16x + 64 = x^2 + 28x + 196.
Minus x^2 from both sides

x^2 + 16x + 64 = 28x+196
Minus 16x 

x^2 + 64 = 12x + 196
Minus 64

x^2 = 12x + 132.
or
x^2-12x-132 = 0.

From there, hopefully you can solve 

So this is a common cause of adding numbers together with large numbers (IDK what its called)
Basically, the equation is: Add the First and last term. (in your case it's 2009). Then divide by 2, and multiply by a number of numbers in that said sequence. 
So how do we how many numbers there are in the sequence? We can use the Explicit function: F(x) = 1+3 (x-1), where f(1)=1. This means that 2008-1 = 2007/3 = 669. This means there are 70 terms. So we have 2009/2) x 70. Or 2009 x 35 which is equal to 70315. 
To double-check my work, I used Wolfram Alpha! 

Welp, this is an easy case of the Chicken and cats problem (is it called that? Idk)
Let's start with calculating what would happen if only 123 adults came. 123 x 7 = 861 dollars. That's only 32 more dollars than they made.

Now, children cost 2 dollars less than adults (that sounds weird, but I'm too lazy to correct it). Therefore, if we divide 32 by 2, we would get 16. 

This means there are 

16 Children

107 adults. 

We double check this: 107 x 7, 16 x 5. 80 + 749 = 829. Thats the answer. 
(I could be wrong, Idk) 

IS it just me or can I not see the picture? I think theirs a picture, but I cannot see it. Can anybody else?

Alright, here we go. Lets count out how many there are for odd starting numbers, and then even starting numbers. 
For odd: The next two have to be even. So 5 possibilities per row, 5 rows, 25 possibilities for 1,3,5,7,9. So 125 for odd
For even its a bit more complicated. Zero doesn't count because that would be 2 digits. So we start with 2. 50 possibilities for that one. x 4. 200. Answers 325. I know I'm not clear but there's the answer. 






 

325. For all you people that don't want to read. 

 Dude. This is probably not what you wanted but here: x=5/9y - 1. 
y= 9/5 +9/5x. Hope that helped. If it didn't make your equation clearer.