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 Mango247, Mango247, Mango247
Let 
and 
be altitudes of an arbitrary scalene triangle 
with orthocenter 
and circumcenter 
. Let 
and 
be the midpoints of sides 
, respectively 
, and 
the intersection point of lines 
and 
. Prove that lines 
and 
are perpendicular.
Liana Topan
and 
be altitudes of an arbitrary scalene triangle 
with orthocenter 
and circumcenter 
. Let 
and 
be the midpoints of sides 
, respectively 
, and 
the intersection point of lines 
and 
. Prove that lines 
and 
are perpendicular.Liana Topan
:
,
:
wrt 
on 
are colinear.
are colinear, the proof is done!
