[/quote]
,
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 
.
board, each square is colored either white or black, with all the squares on the border of the board being black. Additionally, no 
square within the board has all four squares of the same color. Prove that the board contains a 
Y by Adventure10
An acute triangle 
is inscribed in a circle of radius 1 with centre 
all the angles of are greater than 
is the foot of perpendicular from 
to 
is the foot of perpendicular from to 
Similarly,
is the foot of perpendicular from 
to 
is the foot of perpendicular from to 
The lines
and 
intersect at 
The points 
and 
are defined in the same way.
Find the circumradius of triangle
Proposed by F.Bakharev, F.Petrov
is inscribed in a circle of radius 1 with centre 
all the angles of are greater than 
is the foot of perpendicular from 
to 
is the foot of perpendicular from to 
Similarly,
is the foot of perpendicular from 
to 
is the foot of perpendicular from to 
The lines
and 
intersect at 
The points 
and 
are defined in the same way.Find the circumradius of triangle
Proposed by F.Bakharev, F.Petrov
intersect AC at point B' and 
intersect AB at point C'. It is easy to prove that BB' is perpendicular to AC:
then 
and 
,
is an inscribed quadriterial and 
.
is the orthocenter of triangle 
.
and 
are orthocentres of triangles 
and 
respectively.
and 
are congruent triangles 
.
.
as the unit circle centered at 
.
onto 
given by 
.
and 
.
and 
are homothetic with center 
.
is cyclic with 
as diameter and let 
be its circumcenter. Then from the homothety 
.
with 
which is given by 
.
we easily compute 
,
.
.
is the orthocenter of 
.
is the orthocenter it's well-known that the circumradius of 
.
and 
and hence 
,
