[/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 
points of every color there exists a 
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 
.
meets 
.
at 
and 
and 
respectively. The circumcircle of 
meets 
a second time at 
,
meets 
a second time at 
.
passes through the antipodal to 
.
Y by Adventure10
Let 
be a triangle and 
be a circle through the points 
and 
such that intersects 
and 
a second time in the points 
and 
respectively, and being different. Let 
be the centre of . Let 
be the circumcircle of the triangle 
, and let the circumcircle of the triangle intersect also in 
, a point different from
.
Prove that
and
are perpendicular.
be a triangle and 
be a circle through the points 
and 
such that intersects 
and 
a second time in the points 
and 
respectively, and being different. Let 
be the centre of . Let 
be the circumcircle of the triangle 
, and let the circumcircle of the triangle intersect also in 
, a point different from
.Prove that
and
are perpendicular.
