Melody

this question does not make sense.

Attn: mworkhard,

I know you are a valuable member of the forum and maybe I should keep my thoughts to myself......... but really ...

The heading 'quick'   Come on!   Why did you need to sound so demanding?

Then.

"thats wut i got just checking!!"

No "thank you."  no   "I wanted to see if there was a better way to do it"

Did you even look at catmg's or CPhill's working?

You gave your question a point but you did not give a point to either answerer.

I suppose any response is better than no response but you're better than this!

I can see where 2/3 comes from.  I read of the question as the arclength. (not just the distance)

My error.   

I turned a trivial question into something much more complicated.

And please do not post AoPS questions here. 

It is against their code of conduct. 

Plus they claim their questions are copyright so you may be breaking the law. 

MathProblemSolver101

Please do not answer multiple questions like this. This kid is clearly cheating. 

These question still make me thinkg

I do them a bit at a time.

5 people take 27 minutes

so one person must take 27*5=135 minutes

so 18 people will take 135 divided by 18 = 7.5minutes

= 7 minutes 30 sec   just as answer guest said.

Use the formula  

Guest's answers are not correct.

I will start with an equilateral triangle inscribed in a unit circle (radius=1)

Using cosine rule it can be determined that the side length of the triangle is sqrt3

Choose any first point and call it A

The second point can be any point up to sqrt3 units left or right of A so that is a total of 2sqrt3 units

the circumference is 2pi units

So the prob that the 2 points are  closer together  than the sides of the triangle is     

\(\frac{2\sqrt3}{2\pi}=\frac{\sqrt3}{\pi}\)

Say that the first one does happen.  What are the probabilities that the others 2 won't happen?  (Do them seperately)

Multiply the three probs together.

Then do the same for the second one.  etc

when you have done all three, add your answers together.

This is a contour probability map.

The shaded region is the desired region.  

The prob is the area of it over the area of the entire region.