[/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 
and 
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 a tetrahedron 
and 
be the circumcenters of triangles 
respectively. Denote planes 
be the planes which pass through 
and perpendicular to 
respectively. Prove that these planes have a common point called 
If 
is the center of the circumsphere of the tetrahedron, must this tetrahedron be regular?
and 
be the circumcenters of triangles 
respectively. Denote planes 
be the planes which pass through 
and perpendicular to 
respectively. Prove that these planes have a common point called 
If 
is the center of the circumsphere of the tetrahedron, must this tetrahedron be regular?
