AoPS Academy
be a positive integer and let 
be distinct integers in the set 
such that divides 
for 
. Prove that does not divide 
on 
without derivative.
of positive integers is called cardinal if contains the integer 
where denotes the number of distinct elements in . Let 
be a function from the set of positive integers to itself such that for any cardinal set , the set 
is also cardinal. Here denotes the set of all integers that can be expressed as 
where 
. Find all possible values of 
be a parallelogram and point 
be the midpoint of ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
so that the quadrilateral 
is cyclic. If 
is the point of intersection of the lines 
and 
, and 
, then prove that the ray 
is the angle bisector of 
if and only if 
.
be a parallelogram and point be the midpoint of so that the quadrilateral is cyclic. If is the point of intersection of the lines and , and , then prove that the ray is the angle bisector of if and only if .
so 
is an isosceles trapezoid with 
. Hence triangle 
is isosceles and bisector of angle is perpendicular to 
. By ASA 
thus 
. Let 
be the midpoint of 
. We have 
is bisector iff 
such that 
