SOLUTION: Five points lie on a straight line . Alex finds the distance between every pair of points. He obtains in increasing order 2,5,6,8,9,K,15,17,20 and 22. What is K ?

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Question 1069909: Five points lie on a straight line . Alex finds the distance between every pair of points. He obtains in increasing order 2,5,6,8,9,K,15,17,20 and 22. What is K ?
Answer by ikleyn(52908) About Me  (Show Source):
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Answer. K = 14.

Solution 

1.  Let us introduce a conception of segments of the FIRST level.
    They are elementary segments between the given points; segments that do not contain the given points inside.

       It is clear that the segments of the length 2, 5 and 6 are elementary segments   
       (since they are shortest, from one side; and none of them has the length equal to the sum of the two others).

       It is not so clear if the segment of the length 8 is elementary, since 8 = 2+6.
       But the segment of the length 9 is definitely/certainly elementary segment: 9 is not the sum of numbers 2, 5 and 6 in any combination.

       The additional confirmation for it is the fact that 2 + 5 + 6 + 9 = 22 is the ENTIRE segment (maximal distance). 

       After that it is clear that the segment 8 is the combination 8 = 2+6 and is not elementary.

       Moreover, looking in the given sequence of lengths, you can conclude that the segments of the lengths 2 and 5 are the ending segments 
       and the segments 20 = 22-2 and 17 = 22-5 are obtained after removing the ending segments "2" and "5" from the entire 
       "biggest" combined segment of the length 22.

       Thus we have 4 and only 4 elementary segments of the first level.
       They are "2", "5", "6" and "9"; of them, "2" and "5" are ending segments, while "6" and "9" are interior segments.

       And since 2 + 6 = 8,  the segment "6" is the interior segment adjacent to "2".


2.  The segments "17" and "20" are the segments of the THIRD level:
       they obtained from the entire BIG "22" segments after removing "2" and "5" elementary ending segments, respectively.
       Each segment "17" and "20" is the union of three elementary segments of the level 1.


3.  Finally, we have three segments of the SECOND level, that are pairwise unions of the elementary segments, 
    namely ("2"+"6"); ("6"+"9"), and ("9"+"5"). 

    Of them, only 9+5 = 14 is an appropriate candidate for K.

Answer. K = 14.


Solved.