[/quote]
,
be a finite set of cells in a square grid. On each cell of 
an acute triangle inscribed in a circle with its center 
.
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 
.
and 
be on sides 
and 
respectively in 
such that 
.
be the circumcenter of 
and let 
.
Y by CZRorz, Adventure10, Mango247, lian_the_noob12
Let 
be an acute angled triangle whose shortest side is 
. Consider a variable point 
on the side , and let 
and 
be points on 
and 
, respectively, such that 
and 
. Prove that, as traces , the circumcircle of the triangle 
passes through a fixed point.
be an acute angled triangle whose shortest side is 
. Consider a variable point 
on the side , and let 
and 
be points on 
and 
, respectively, such that 
and 
. Prove that, as traces , the circumcircle of the triangle 
passes through a fixed point.
.
be the bisector of 
with 
on 
Clearly, 
is the circumcenter of 
are kites. 
bisects the 
respectively. 
is the incenter.
,
is isosceles and 
is 
angle bisector, then 
so 
and 
Similarly, 
and 
Hence, 
so 
Note that 
and 
Hence, 
.
so 
is cyclic so all 
goes through 
,
are concyclic
and 
are kite which implies 
.
is the 
a
.
which is fixed therefore by using 
.
wrt ![\[ f(\bullet)=P(\bullet, (ADE))-P(\bullet, (ABC)) \]](http://latex.artofproblemsolving.com/5/f/1/5f1b4debfedc233c68d96966a28a06b603fe139e.png)
It is known that 
.
is fixed. By the linearity we have
which is fixed.
The fixed point is 
.
, 
because 
. Likewise 
.
.
and 
.
the problem will end. 
because 
.
because it is the central angle 
