[/quote]
,
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 
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)
.
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 
meets 
a second time at 
meets 
a second time at 
passes through the antipodal to 
Y by Adventure10, Mango247
The two circles 
and 
intersect at 
and 
. The common tangent that's on the same side as , intersects the circles at 
and 
,respectively. Let 
be the second intersection with of the tangent to at , and let 
be the second intersection with of the tangent to at . Let 
be the intersection of 
and 
, and let 
be the intersection of 
and 
. Let 
be the image of under point reflection with respect to the midpoint of 
. Prove that 
is a cyclic hexagon.
and 
intersect at 
and 
. The common tangent that's on the same side as , intersects the circles at 
and 
,respectively. Let 
be the second intersection with of the tangent to at , and let 
be the second intersection with of the tangent to at . Let 
be the intersection of 
and 
, and let 
be the intersection of 
and 
. Let 
be the image of under point reflection with respect to the midpoint of 
. Prove that 
is a cyclic hexagon.This post has been edited 1 time. Last edited by parmenides51, Aug 3, 2019, 7:53 AM
cyclic and 
cyclic
