Melody

You are right, So I did.     

Thanks for picking that up.  :)

I can do it a third way, although to be honest I do not know what Vieta's formula is

Sum of the roots in any polynomial is  = -b/a

In this case that is  -12/1 = -12

let one root be T the other is twice the size so   

T+2T=-12

3T=-12

T= -4

So the roots are  -4 and -8  so k is 32

If I understand the problem properly then I think that there are 4 possibilities for each combination of 4 colours/

There are 6C4 = 15 ways to choose those 4 colours

So I get 4*15 = 60 

There are  9*10*10*10 = 9000  4 digit numbers

There are  9*9*8*7 = 4536 that do not have repeating digits.

SO there are 9000-4536 = 4464 that do

x = 4464/9000*100 = 49.6

Gore, I know you are very capable but on this occasion you did not read the question properly.

\(S_n=\frac{n}{2}(2a+(n-1)d)\\~\\ 10^{10}=\frac{100}{2}(2a+(100-1)*2)\\~\\ 10^{10}=\frac{100}{2}(2a+99*2)\\~\\ 10^{10}=100(a+99)\\~\\ 10^{8}=a+99\\~\\ a=10^{8}-99\\~\\\)

Check

\(S_n=\frac{n}{2}(2a+(n-1)d)\\~\\ S_{100}=\frac{100}{2}(2(10^8-99)+99*2)\\~\\ S_{100}=100(10^8-99+99)\\~\\ S_{100}=100(10^8)\\~\\ S_{100}=100^{10} \)

Of course this is not a great check because I have used exactly the same formula as I did when i worked it out.

S_n=\frac{n}{2}(2a+(n-1)d)\\~\\

10^{10}=\frac{100}{2}(2a+(100-1)*2)\\~\\

10^{10}=\frac{100}{2}(2a+99*2)\\~\\

10^{10}=100(a+99)\\~\\

10^{8}=a+99\\~\\

a=10^{8}-99\\~\\

I have no idea what uniformly at random means ?   

Random is self explanatory but uniformly is not.

This is how Gor.. came up with this answer.

9/10 < a/b < 10/11

so

18/20 < a/b < 20/22

There is a fair bet that 19/21 will be between the 2.  (Maybe it is certainty, but I wouldn't want to say that without proving it)

Check by using decimals

9/10 = 0.9

10/11 = 0.909090....

19/21 = 0.90476

And it is between  the 2 as predicted.

The smallest value of b is 21

Well everyone hs there own methods for memorising things.

For these ones i would not worry too much about the letters as they change according to how a triangle is labeled. 

You do have to be careful about labeling Angles with capital letters and the opposite side with the little letter equivalent.

The sine rule is just

Any side over the sine of its opposite angle, is equal to any other side over the sine of its opposit angle.

\(\frac{a}{SinA}=\frac{b}{SinB}=\frac{c}{Sin C}\)

The Cosine rule

I am discussing how it is used when you are looking for a side. 

                     A lot of people would use a rearranged version  when they are looking for an angle

The cosine rule starts like pythagoras's theorum  

 The side you want squared^2 = the sum of the squares on the other 2 sides     eg   \(a^2=b^2+c^2\)

Then you take away 2* product of the sides you have * Cos (the angle opposite the one you have.

\(a^2=b^2+c^2-2bc\;cosA\)

That probably didn't sound that simple, so lets discuss it more.

The side letter at the front is the same as the angle letter at the end.

   So you start with the side you want to find and end with the cos of its opposite angle    \(a^2= \dots\dots cosA\)

The beginning looks like pythagoras's theorum.     \(a^2=b^2+c^2\dots \dots cosA\)

     then it is take away.. there is no more squares but we use the 2 again.  time the two sides that we know. -2bc

                                                                              \(a^2=b^2+c^2-2bc\; cosA\)

EXAMPLE

Say your triangle was called PTR and you needed side PR  

If you draw it you will see that side PR is opposite T  so you should rename it   t

The formula would be

\(t^2=p^2+r^2-2prCosT\)

Maybe that will help, maybe not, you decide :)