such that ![\[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]](http://latex.artofproblemsolving.com/0/6/2/0623b7c328af6d088cfc5dd288afe3e17a94ff29.png)
for all reals 
and 
.
and 
be two sequences of integers such that 
and 
for all 
.
divides 
for an infinity of integers 
such that 
for all 
are coprime. Consider the sequence 
given by 
.
such that 
is even and 
is an integer. Prove that the numerator in its simplest form of 
is an odd integer greater than 
.
be an integer and let 
be 
.
such that the n subsets 
are also disjoint.
s
.
be reals with 
a
.
is the incentre and point 
is defined such that 
and 
is the perpendicular bisector of 
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
be non-negative reals such that 
.
people seated around a circular table, and 
cookies
be one of these people. Find the least 
and 
.
.
Y by
MO Space City plans to construct 
space stations, with a unidirectional pipeline connecting every pair of stations. A station directly reachable from station P without passing through any other station is called a directly reachable station of P. The number of stations jointly directly reachable by the station pair 
is to be examined. The plan requires that all station pairs have the same number of jointly directly reachable stations.
(1) Calculate the number of unidirectional cyclic triangles in the space city constructed according to this requirement. (If there are unidirectional pipelines among three space stations A, B, C forming
, then triangle ABC is called a unidirectional cyclic triangle.)
(2) Can a space city with stations meeting the above planning requirements be constructed for infinitely many integers 
?
space stations, with a unidirectional pipeline connecting every pair of stations. A station directly reachable from station P without passing through any other station is called a directly reachable station of P. The number of stations jointly directly reachable by the station pair 
is to be examined. The plan requires that all station pairs have the same number of jointly directly reachable stations.(1) Calculate the number of unidirectional cyclic triangles in the space city constructed according to this requirement. (If there are unidirectional pipelines among three space stations A, B, C forming
, then triangle ABC is called a unidirectional cyclic triangle.)(2) Can a space city with
stations meeting the above planning requirements be constructed for infinitely many integers 
?
