[/quote]
,
,
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 
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 
be the midpoint of side 
,
and 
,
and 
intersect at a point 
,
is perpendicular to line 
such that ![\[f(f(x)) + xf(xy) = x + f(y)\]](http://latex.artofproblemsolving.com/a/6/f/a6f84dd53eec985b90bad3df6580eeaf29d73195.png)
for all positive real numbers 
and 
.
,
with center 
and radius 
intersects the circle 
and 
.
.
such that for any real value of 
we have:
and 
are points, such that 
,
= 
and 
and 
passes through the fix point.
Consider an 
-
in the plane. An eligible set is a non-empty set of the form 
where 
determine the smallest possible number of eligible subsets 
such that :
Y by
Let 
be an acute triangle with orthocenter 
and circumcenter 
. Let 
be the intersection point of 
with 
and 
be the intersection point of 
with 
. It is known that 
. Prove that 
is a parallelogram.
be an acute triangle with orthocenter 
and circumcenter 
. Let 
be the intersection point of 
with 
and 
be the intersection point of 
with 
. It is known that 
. Prove that 
is a parallelogram.This post has been edited 1 time. Last edited by parmenides51, Oct 19, 2022, 5:39 AM
