SOLUTION: N lattice points in coordinate 8-space are chosen at random. The line segments joining all pairs of these N points are drawn. What is the smallest value of N that guarantees that t

Each lattice point  x = (, , , . . . , )  is the set of 8 (eight) integer numbers, - its coordinates, 
that are positive, zero or negative integer numbers.


The line segments, joining all pairs of these lattice points, are vectors, whose components are the differences of the 
lattice points' coordinates.


For each such a joining segment/vector, its midpoint is a lattice point if and only if ALL 8 components of this segment/vector 
are EVEN numbers.


As I just said above, each component of the joint segment/vector is the difference of the coordinates of the original 
lattice points.


Therefore, for every joining segment/vector, its midpoint is a lattice point if and only if for the respective pair of the lattice points 
all 8 their coordinates have the same parity as integer numbers, i.e. in each of 8 positions they are either EVEN or ODD integer 
numbers simultaneously.



Next, it is clear, that in any set of  vectors with integer components in 8-space , at least two vectors have identical parity 
of integer components in all 8 positions.


    +-------------------------------------------------+
    |    It follows from the "pigeonhole principle".  |
    +-------------------------------------------------+



It leads us to the conclusion that if    (the number of all joining segments)  is greater than   = 257,

then there are at least two vectors that have identical parity of integer components in all 8 positions.


Therefore, the minimal number N the problem asks for, is N satisfying inequality


     >= 257,   or


    N*(N-1) >= 2*257 = 514.


And since   = 22.67,  the final inequality and the ANSWER  is N >= 24.


ANSWER.  The smallest value of N that guarantees that the midpoint of at least one of the resulting line segments is itself a lattice point, is 24.