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and 
be positive integers with 
. In a group of people, each one or always
, what is the parity of people who speak to is a subset of size 
of the set of people. The answer can only
” or “
”. and is it possible to determine which people speak the truth and
be pairwise different positive real numbers such that![\[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\]](http://latex.artofproblemsolving.com/8/a/e/8ae0bd1f55eb2fb0b7209b62739a9b98f462ee56.png)
is an integer for every 
Prove that 
of positive integers that satisfy
. Find the minimum and maximum values of 
+ 
+ 
. Let be a subset of 
such that no sum of three distinct elements of is divisible by 
. What is the maximum value of 
.
obeys the recurrence 
for any integers 
. Additionally, 
and 
. Let 
can be expressed as 
for two relatively prime positive integers 
and . Determine the value of 
.
be a rhombus, and E be the midpoint of side CD. Let F be a point on BE such that
angles and 
are equal 
. On the sides 
and 
points 
and 
are selected so that 
. It turned out that the line 
is parallel to the angle bisector of the angle 
. Find the angle 
.
angles and are equal . On the sides and points and are selected so that . It turned out that the line is parallel to the angle bisector of the angle . Find the angle .
through parallel to 
. Let intersect 
at 
. Notice, that has to be bisector of angle and, as a consequence, of angle 
in 
. Thus, triangles 
and 
are equal by angles and common side. So 
is parellelogram 
