[/quote]
,
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)
.
be reals such that 
and 
.
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 
,
,
is __
prove that![$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$](http://latex.artofproblemsolving.com/6/a/6/6a657cb647e279b0745e3189c3f9d54aa25bf976.png)
satisfying 
is inscribed in circle 
.
denote its incenter and 
its 
-
denote the midpoint of arc 
.
meets 
.
at 
and 
at 
and 
respectively. The circumcircle of 
meets 
a second time at 
,
meets 
a second time at 
.
passes through the antipodal to 
Y by Adventure10
Let 
and 
be two spheres with distinct radii that touch externally. The spheres lie inside a cone 
, and each sphere touches the cone in a full circle. Inside the cone there are 
additional solid spheres arranged in a ring in such a way that each solid sphere touches the cone , both of the spheres and externally, as well as the two neighboring solid spheres. What are the possible values of ?
Proposed by Iceland.
and 
be two spheres with distinct radii that touch externally. The spheres lie inside a cone 
, and each sphere touches the cone in a full circle. Inside the cone there are 
additional solid spheres arranged in a ring in such a way that each solid sphere touches the cone , both of the spheres and externally, as well as the two neighboring solid spheres. What are the possible values of ?Proposed by Iceland.
.
,
of 
or 
.
of the cone, we have 
.
by 
(this is the corresponding angle intervening between 
denote the distance of the center of ![\[R=(b+r)\sin(\phi+\psi)=2\frac{(a-b)\sqrt{br}+(b-r)\sqrt{ab}}{a+b}=\frac{2ab(a+b+\sqrt{ab})}{(a+b)(\sqrt{a}+\sqrt{b})^2}\]](http://latex.artofproblemsolving.com/4/7/1/471516ca8a4302ec12884cbcd9f3cecd6efc58fc.png)
where 
.
with values between 
and 
.
,
.
(for 
)
