[/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 
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
In the circle circumscribed to the triangle 
, the respective midpoints 
are considered of arcs 
, 
and 
that do not contain the third vertex. Lines 
and 
intersect BC at points 
and 
, respectively. and 
intersect at 
and 
,respectively. 
and intersect
at 
and 
, respectively. Prove that the proportions are verified
and that lines 
, 
and 
intersect at the center of the circle inscribed in triangle .
Note: As usual
and 
are the lengths of the sides 
and of the triangle, respectively.
, the respective midpoints 
are considered of arcs 
, 
and 
that do not contain the third vertex. Lines 
and 
intersect BC at points 
and 
, respectively. and 
intersect at 
and 
,respectively. 
and intersect
at 
and 
, respectively. Prove that the proportions are verified
and that lines 
, 
and 
intersect at the center of the circle inscribed in triangle .Note: As usual
and 
are the lengths of the sides 
and of the triangle, respectively.
