[/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 
-
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, Mango247
Points 
and 
are chosen on a circle in such a way that the point 
of the intersection of the lines 
and 
lies outside the circle and the point 
of intersection of the lines 
and 
lies inside the circle. Let points 
and 
lie on chords and respectively and 
denotes point of intersection of the lines 
and 
. Prove that 
.
and 
are chosen on a circle in such a way that the point 
of the intersection of the lines 
and 
lies outside the circle and the point 
of intersection of the lines 
and 
lies inside the circle. Let points 
and 
lie on chords and respectively and 
denotes point of intersection of the lines 
and 
. Prove that 
.
to be distinct (otherwise: guess what will happen 
)
.
and 
are similar, so![\[ \frac {SC}{SB} = \frac {AC}{BD}.
\]](http://latex.artofproblemsolving.com/6/8/7/687af471fd13b32c0fd54340146862bcf359c6cd.png)
and 
respectively, we obtain![\[ \frac {TM}{TN} \cdot\frac {KN}{KM} = \frac {\sin\angle NTK}{\sin\angle MTK}.
\]](http://latex.artofproblemsolving.com/9/5/1/95132a5e5f12e6fe2ace1be5b88862f25e943688.png)
