[/quote]
,
,
be the set of all positive integers not exceeding 
that are relatively prime to ![\[
P_n(x) = \sum_{k \in S_n} x^{k - 1}
\]](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 
at 
and 
respectively. The circumcircle of 
meets 
a second time at 
,
meets 
a second time at 
.
passes through the antipodal to 
be a sequence defined by 
.
if 
.
Y by Mango247
Given a circle of radius 
, four tangent lines are drawn intersecting (in pairs) at 
points 
such that 
is a rhombus with 
. A line 
is drawn, tangent to the circumference and intersects the sides 
and CD of the rhombus at points 
and 
respectively. Find the area of the triangle 
and show that it is the same for any choice of line .
, four tangent lines are drawn intersecting (in pairs) at 
points 
such that 
is a rhombus with 
. A line 
is drawn, tangent to the circumference and intersects the sides 
and CD of the rhombus at points 
and 
respectively. Find the area of the triangle 
and show that it is the same for any choice of line .
.
on the circle 
.
cuts 
in the point 
.
,
