Blue and red dots on a line

[Exploited]

You have a line made from blue and red dots. Please proof that you can find three equally colored dots on the line, that one of them is exactly in the middle of the segment, made by the others.

Edit: Easier explanation - you have line from blue and red dots. Proof that you can find three blue (or red) dots A, B and C, that B is a middle of the segment AC

Note: The dots are absolutely randomly placed and random numbers - blue can be more than reds and etc... Totally random

[Exploited]

not popular problem here But here is the proof:

Let we take two random blue dots (red will be "o" and blue will be "x") :

-----------------x---------------x------------------------------------

if their middle is blue - we have a proof => we will say NO - the middle is red:

-----------------x-------o-------x--------------------

Let we look to the dot in the right of the third one - it must be blue (or we will proove):

-----------------x-------o-------x-------x------------

Now let we look in the following dot - D:

---------D-------x-------o-------x-------x------------
             1          2         3          4         5

If D is blue then 3 (red) will be middle of two blue dots (1 and 5) => prooved
If D is red then 2 (blue) will be middle of two red dots (1 and 3) => prooved

=> ... prooved