where 
where 
.
cannot be a perfect cube of a positive integer.
is given. Let 
be set of all 
segments of real line of type ![$[i, j]$](http://latex.artofproblemsolving.com/a/6/4/a644edd67caa59eaded765c417b6d9822f782ec2.png)
,
and 
are integers, 
.
is said to be valuable if the intersection of any two segments from 
is either empty, or is a segment of nonzero length belonging to 
with real coefficients obeying![\[P(x) P(x+1) = P(x^2 + x + 1)\]](http://latex.artofproblemsolving.com/b/e/f/beff3420fa8394f4f98c031d81eea6a5a3b8d28c.png)
for all real numbers 
.
Y by
In connection with the formation of a stable government, the President invited all 
Members of Parliament to three separate consultations, where each member participated in exactly one consultation, and at each consultation there has been at least one member present. Discussions between pairs of members are to take place to discuss the consultations. Is it possible for these discussions to occur in such a way that there exists a non-negative integer 
, such that for every two members who participated in different consultations, there are exactly members who participated in the remaining consultation, with whom each of the two members has a conversation, and exactly members who participated in the remaining consultation, with whom neither of the two has a conversation? If yes, then find all possible values of .
Members of Parliament to three separate consultations, where each member participated in exactly one consultation, and at each consultation there has been at least one member present. Discussions between pairs of members are to take place to discuss the consultations. Is it possible for these discussions to occur in such a way that there exists a non-negative integer 
, such that for every two members who participated in different consultations, there are exactly members who participated in the remaining consultation, with whom each of the two members has a conversation, and exactly members who participated in the remaining consultation, with whom neither of the two has a conversation? If yes, then find all possible values of .
