[/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
On a plane are given three non-collinear points 
. We are given a disk of diameter different from that of the circle passing through large enough to cover all three points. Construct the fourth vertex of the parallelogram 
using only this disk (The disk is to be used as a circular ruler, for constructing a circle passing through two given points).
. We are given a disk of diameter different from that of the circle passing through large enough to cover all three points. Construct the fourth vertex of the parallelogram 
using only this disk (The disk is to be used as a circular ruler, for constructing a circle passing through two given points).
