[/quote]
,
.
that satisfies![\[ f(x+yf(x))f(y)=f(x)(1+yf(y+1)) \]](http://latex.artofproblemsolving.com/f/7/a/f7a138fd2dc76f034525491a10f62f1e83de294d.png)
for all 
.
,
be positive integers such that 
is a multiple of 
for all positive integers 
.
with 
is called historic if 
.
satisfying 
for all pairs of positive reals 
and 
.
denotes the set of all positive reals.
be positive real numbers such that : 
. Prove that : 
L
be positive reals numbers such that 
.
satisfying: 
,
and the function 
defined by![\[f(n)=\left\lfloor \frac{an+b}{cn+d}\right\rfloor\text{ for all } n\in\mathbb{Z}_{\ge 0}.\]](http://latex.artofproblemsolving.com/8/f/b/8fb372df4b44c3d7c0690b310b12c684134fee74.png)
Prove that the following are equivalent:
is surjective;
,
and 
.
and 
be on sides 
respectively in 
such that 
.
be the circumcenter of 
and let 
.
,
is the set of real numbers, such that
be the n-th prime number in increasing order for 
.
satisfying 
for all 
Y by Adventure10, Mango247
Let 
be a non-right-angled triangle and let 
be its orthocenter. Let 
be the midpoints of the sides 
, 
, 
respectively. Let 
, 
, 
be the symmetrical points of with respect to 
, 
and 
respectively, and let 
, 
, 
be the orthocenters of the triangles 
, 
and 
respectively. Prove that:
a) triangles and 
have the same centroid;
b) the centroids of the triangles
, 
, 
form a triangle similar with .
be a non-right-angled triangle and let 
be its orthocenter. Let 
be the midpoints of the sides 
, 
, 
respectively. Let 
, 
, 
be the symmetrical points of with respect to 
, 
and 
respectively, and let 
, 
, 
be the orthocenters of the triangles 
, 
and 
respectively. Prove that:a) triangles
and 
have the same centroid;b) the centroids of the triangles
, 
, 
form a triangle similar with .
are the midpoints of 
respectively. Hence 
have the same centroid.
be the centroids of 
respectively. It is trivial that 
,
and 
are collinear, and 
.
.
,
.
