[/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 
.
be on sides 
and 
respectively in 
such that 
.
and let 
.
Y by Adventure10, Mango247
Let 
and 
be two circles, of centers 
and 
respectively, intersecting at 
and 
. The straight line 
is the common tangent to and , closer to . Points 
and 
are the respective points of contact of with and 
the point diametrically opposite and 
the intersection point of the line 
with the line perpendicular to line 
drawn by . Show that 
and 
are collinear.
and 
be two circles, of centers 
and 
respectively, intersecting at 
and 
. The straight line 
is the common tangent to and , closer to . Points 
and 
are the respective points of contact of with and 
the point diametrically opposite and 
the intersection point of the line 
with the line perpendicular to line 
drawn by . Show that 
and 
are collinear.This post has been edited 1 time. Last edited by parmenides51, Jun 21, 2022, 1:41 AM
