[/quote]
,
be a finite set of cells in a square grid. On each cell of 
.
and 
the perpendiculars in 
respectively, where 
,
and 
.
be the circumcircles of triangles 
.
and 
such that 
and 
.
is a perfect square.
.
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 
.
such that 
.
be the circumcenter of 
and let 
.
Y by Adventure10, Mango247
Let 
be a triangle and 
be the angle bisector of 
(
). The point 
is on the segment and 
is the midpoint of the side 
. The point 
is on the line connecting and such that is the midpoint of 
. Define 
and 
as the intersections of 
, 
, and 
, 
. Prove that 
.
be a triangle and 
be the angle bisector of 
(
). The point 
is on the segment and 
is the midpoint of the side 
. The point 
is on the line connecting and such that is the midpoint of 
. Define 
and 
as the intersections of 
, 
, and 
, 
. Prove that 
.
iff the circumradii of triangles BQE and CQD are equal, so maybe it can be proven by using the other sides and angles and trig
respectively.
Hence, it suffices to show that 
is parallelogram 
.
,
.
.
,
, remark that 
since 
have the same midpoint thus 
but by Ceva's we deduces 
therefore the result follows .
,
,
and 
