[/quote]
,
attaining all positive values and positive numbers 
are given. Numbers 
satisfy
for every 
.
satisfying 
.
?
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 
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)
Y by Adventure10, Mango247
Inscribed in a triangle 
with the center of the circle 
touch the sides 
and 
at points 
and 
, respectively. The point 
divides the segment 
in a 3:1 ratio, measured from . 
- the midpoint of . Prove that the points 
lie on a circle, if you know that 
.
with the center of the circle 
touch the sides 
and 
at points 
and 
, respectively. The point 
divides the segment 
in a 3:1 ratio, measured from . 
- the midpoint of . Prove that the points 
lie on a circle, if you know that 
.
cuts 
at its midpoint 
.
,
is cyclic, iff 
.
![${\color[rgb]{1,1,1}.}\ \square \square \square$](http://latex.artofproblemsolving.com/5/0/7/507414e7f169c6f2ea193548fb11d6f90b09d230.png)
