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, where 
.
be a positive integer and 
be the remainder when 
is divided by . for which the sequence 
is eventually periodic.
such that 
to 
are to be put into two identical boxes so that each box contains at least one ball and the greatest common divisor of the product of the labels of all the balls in one box and the product of the labels of all the balls in the other box is . Determine the number of ways that this can be done.
with real coefficients such that for all real 
, 
.
and 
such that the equation 
at least one solution 
where and 
are positive integers.
is inscribed in a circle. Show that the line connecting the midpoints of the arcs 
and 
and the line connecting the midpoints of the arcs 
and 
are perpendicular.
positive integers into 
pairs is called square-free if the product of numbers in each pair is not a perfect square.Prove that if for distinct positive integers, there exists one square-free partition, then there exists at least 
square-free partitions.
with 
intersect at 
, and 
and 
are the circumcenter and incenter of 
, respectively. Prove that if 
then 
and are perpendicular
be real numbers such that 
and 
, for all 
. Prove that 
for all 
and ![$\big [ \frac{2^n}{n} ]$](http://latex.artofproblemsolving.com/8/6/e/86ea8d32d5a9dfd2562c829a3baf7f65ef2356b0.png)
is a power of 
, then is a power of .
such that ![$2^k=\left[\frac{2^n}{n}\right ]$](http://latex.artofproblemsolving.com/d/a/1/da1f74e3a6ac06d139c7d7707214c812ade55795.png)
. If 
there is nothing to prove, so let us assume 
. One can easily check inequality 
, so 
or 
.
and 
so is situated between two consequtive integers.It is possible iff 
, q.e.d.
