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 
where 
.
cannot be a perfect cube of a positive integer.
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 
.
and![\[(x-y)[f(x)+f(y)]\leqslant f(x^2-y^2), \forall x,y \in \mathbb{R}\]](http://latex.artofproblemsolving.com/8/0/b/80bb1d7a992c9683a0c2a07b2febe6271f4e60f5.png)
be rational numbers such that 
is rational.
.
so division by 
is possible and hence
and both 
and 
Y by
In a school with double schooling, in the morning the language teacher divided the students into 
groups for an activity. In the afternoon, the math teacher divided the same students into 
groups for another activity. A student is considered special if the group they belonged to in the afternoon is smaller than the group they belonged to in the morning. Find the minimum number of special students that can exist in the school.
Note: Each group has at least one student.
groups for an activity. In the afternoon, the math teacher divided the same students into 
groups for another activity. A student is considered special if the group they belonged to in the afternoon is smaller than the group they belonged to in the morning. Find the minimum number of special students that can exist in the school.Note: Each group has at least one student.
.
groups have a random grouping. Let then the 
students, let each of these 
.
,
.
,
.
