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, Mango247
ABC is a triangle and 
the circumcircle of ABC. 
, 
, 
are the midpoints of the sides BC, AC and AB. D is the feet of the altitude from A and 
is the projection of on 
. G is the centroid of ABC. 
is the circumference which passes through and is tangent to in P (different from A). Show that:
1) and the tangents to in A and P are concurrent;
2) P,D,G, are collinear.
the circumcircle of ABC. 
, 
, 
are the midpoints of the sides BC, AC and AB. D is the feet of the altitude from A and 
is the projection of on 
. G is the centroid of ABC. 
is the circumference which passes through and is tangent to in P (different from A). Show that:1)
and the tangents to in A and P are concurrent;2) P,D,G,
are collinear.
is image of 
under the homothety with center 
and scale 
,
,
,
,
is foot 
.
and any radius.
and 
,
at 
are basically that pairwise radical axex of the three circles 
is on the perpendicular bisector of 
,
,
is 
(
be the circumcenter of 
.
,
are concyclic, which means that 
(
a
,
are collinear, hence 
are collinear
be the point on 
,
.
