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, 
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 
. Let 
be the circumcenter of 
and let 
. Prove that 
, where 
is the set of real numbers, such that
is a parallelogram in which angle 
is acute. Points 
lie on one circle in exactly this order. Lines 
and 
intersect in 
. Point 
is the circumcenter of the triangle 
. Prove that if 
then the lines 
and 
are perpendicular.
is a parallelogram in which angle is acute. Points lie on one circle in exactly this order. Lines and intersect in . Point is the circumcenter of the triangle . Prove that if then the lines and are perpendicular.
. 
determine a a cyclic trapezoid, hence 
, i.e. 
is isosceles, so, obviously, we need to prove 
are con-cyclic, which is obvious: 
, done.
and 
.
.
and 
.
.
and using law of sines we must prove
