Melody

Jeff, stop asking AoPS questions.  You know it is cheating and it gives our forum a bad name.

You are completely out of your depth KitSoundwave

This is not written in Latex and LaTex works just fine.

What you have guessed makes no sense because you have no idea what you are talking about.

I like your enthusiasm but please stay within your knowledge base.

What is under the square root?

I think there are likely other errors in there as well.

Please check your question carefully.

Where is your working?

Your pic is permanently blocked.

I do not know why but maybe you do.

Estimated to what?  You have not asked a question.

2 parts of the bag make 18 marbles.

So one part must be 9 marbles

3 parts of the bag are blue so there must be 9*3 =27 blue marbles.

Next time please post only one question at a time.

If I had seen it before the other answerers this question would have been deleted.

But it appears that you tried to learn from their answers so it is probably just as well they saw it before me.  

Next time, just one question though.

Have you tried drawing the picture.  

This question really is not very hard.

Once you have the pic you can use Pythagoras's theorem.

Thanks textot  

Here is another method:

\(f(100)=100^2-100+2010\)

\(f(101)=101^2-101+2010\\ f(101)=(100+1)^2-101+2010\\ f(101)=100^2+200+1-101+2010\\ f(101)=100^2-100+200+2010\\ f(101)=f(100)+200\\\)

Now my problem has changed to, what is the highest common factor of

  \(100^2-100+2010 \qquad and \qquad 200\)

10 goes into both so I will divide by 10

\(1000-10+201 \qquad and \qquad 20\)

The last digit of the first one is 1.   Nothing ending in 1 will have any common factors with 20, (it would have to end in a multiple of 2 or 5)

So

The hight common factor is 10.