[/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 
and 
respectively. The circumcircle of 
meets 
a second time at 
,
meets 
a second time at 
.
passes through the antipodal to 
.
Y by Adventure10
Kilua and Ndoti play the following game in a square 
: Kilua chooses one of the sides of the square and draws a point 
at this side. Ndoti chooses one of the other three sides and draws a point Y. Kilua chooses another side that hasn't been chosen and draws a point Z. Finally, Ndoti chooses the last side that hasn't been chosen yet and draws a point W. Each one of the players can draw his point at a vertex of , but they have to choose the side of the square that is going to be used to do that. For example, if Kilua chooses 
, he can draws at the point 
and it doesn't impede Ndoti of choosing 
. A vertex cannot de chosen twice. Kilua wins if the area of the convex quadrilateral formed by , 
, 
, and 
is greater or equal than a half of the area of . Otherwise, Ndoti wins. Which player has a winning strategy? How can he play?
: Kilua chooses one of the sides of the square and draws a point 
at this side. Ndoti chooses one of the other three sides and draws a point Y. Kilua chooses another side that hasn't been chosen and draws a point Z. Finally, Ndoti chooses the last side that hasn't been chosen yet and draws a point W. Each one of the players can draw his point at a vertex of , but they have to choose the side of the square that is going to be used to do that. For example, if Kilua chooses 
, he can draws at the point 
and it doesn't impede Ndoti of choosing 
. A vertex cannot de chosen twice. Kilua wins if the area of the convex quadrilateral formed by , 
, 
, and 
is greater or equal than a half of the area of . Otherwise, Ndoti wins. Which player has a winning strategy? How can he play?This post has been edited 2 times. Last edited by Kowalks, Dec 29, 2014, 2:31 PM
.
is a triangle inscribed in the square, of area at most half of that of the square.
.
and 
.
and 
,
.
,
,
,
.
.
