[/quote]
,
be real numbers such that 
.
Let 
.
and 
,
as follows:
where 
are all the positive divisors of 
.
Find all positive integers 
.
Find all positive integers 
.
be a convex quadrilateral. Suppose that the circles with diameters 
and 
intersect at points 
and 
.
and 
.
,
,
attaining all positive values and positive numbers 
are given. Numbers 
satisfy
for every 
.
.
?
be reals such that 
and 
.
and 
points on a plane is 
,
such that: in every set of 
points of a plane, no three of which are collinear, consisting of 
consisting of 
disting positive integers smaller than 
are given. Prove that there exists a positive integer 
that can be written in the form 
,
in at least 
different ways.
,
be the set of all positive integers not exceeding ![\[
P_n(x) = \sum_{k \in S_n} x^{k - 1}
\]](http://latex.artofproblemsolving.com/3/a/c/3ac60cb70baf4dfcf297e9af87480a8ee34ba70c.png)
and a polynomial 
with integer coefficients such that![\[
P_n(x) = (x^{r_n} + 1) Q_n(x).
\]](http://latex.artofproblemsolving.com/7/0/e/70e19a9d3e0a7e97abc7aa08d3f23126c14ea7d0.png)
is irreducible in ![$\mathbb{Z}[x]$](http://latex.artofproblemsolving.com/4/b/7/4b7d768fffbb7a1a6163af392ed348f2d49d8783.png)
.
integers are such that 
is prime and 
.
such that 
.
Y by Adventure10, Mango247
Let 
be a triangle and 
the projections of 
on their opposite sides. Let 
be the projection of 
on 
respectively on 
are defined analogously. Moreover, 
is the intersection of 
with 
the intersection of 
with 
the intersection of 
with 
Show that
and 
are concurrent.
be a triangle and 
the projections of 
on their opposite sides. Let 
be the projection of 
on 
respectively on 
are defined analogously. Moreover, 
is the intersection of 
with 
the intersection of 
with 
the intersection of 
with 
Show that
and 
are concurrent.
.
w
be reflections of 
over 
.
,
and similarly 
are midlines of 
is the medial triangle of the orthic triangle.
and similarly others so 
parallel to 
?
