and 
.
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 
.
Y by Adventure10
Consider a scalene triangle 
and let 
and 
be two variable points such that 
. If 
and the circumcircles of 
and 
meet each other at 
and 
, prove that the reflection of across the midpoint of 
belongs to a fixed line.
Proposed by Liubomir Chiriac, student, Chișinău, Moldova
and let 
and 
be two variable points such that 
. If 
and the circumcircles of 
and 
meet each other at 
and 
, prove that the reflection of across the midpoint of 
belongs to a fixed line.Proposed by Liubomir Chiriac, student, Chișinău, Moldova
.
,
with 
.
.
lies on the line 
.
and 
.
.
,
.
