Suppose that |a-b|+|b-c|+|c-d|+.......+|m-n|+|n-o|+............|x-y|+|y-z|+|z-a|=20. What is the maximum possible value of |a-n|?

Answer 1

Because it is circularly symmetrical, all 26 variables cannot have the same value (otherwise the sum would be zero).

To maximize the value of |a-n|, we need
|a-n|=10 which in the integer domain gives
±(0,-10),±(1,-9),±(2,-8),...±(5,-5),

Answer 2

add: ...and the remaining 24 variables equal to zero.