Valuable subsets of segments in [1;n]
by NO_SQUARES, Apr 3, 2025, 8:34 PM
The integer 
is given. Let 
be set of all 
segments of real line of type ![$[i, j]$](//latex.artofproblemsolving.com/a/6/4/a644edd67caa59eaded765c417b6d9822f782ec2.png)
, where 
and 
are integers, 
. A subset 
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 . Find the number of valuable subsets of set .
is given. Let 
be set of all 
segments of real line of type ![$[i, j]$](http://latex.artofproblemsolving.com/a/6/4/a644edd67caa59eaded765c417b6d9822f782ec2.png)
, where 
and 
are integers, 
. A subset 
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 . Find the number of valuable subsets of set .
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)
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 
.
where 
.
cannot be a perfect cube of a positive integer.
such that:
such that 
where 
,
and 
.
is prime, 
with 
and 
?
be the circumcircle of isosceles triangle 
)
lie on 
respectively such that 
.
to the incircle of 
(different from
,
and 
be three sequences of real numbers which obeys all the following :
Forgetting the third condition (as given above) can lead us to errorenous results.
and both the sequences 
April 2021
February 2021