[/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
Let 
be the perimeter of an acute-angled triangle 
which is not an equilateral triangle. Let 
be a variable points inside the triangle , and let 
be the projections of on the sides 
respectively. Prove
that![\[ 2(AF+BD+CE ) = \ell \]](//latex.artofproblemsolving.com/9/7/4/974ff5c3a9064b29f3b47499d21f630e2c268bae.png)
if and only if is collinear with the incenter and the circumcenter of the triangle .
be the perimeter of an acute-angled triangle 
which is not an equilateral triangle. Let 
be a variable points inside the triangle , and let 
be the projections of on the sides 
respectively. Provethat
![\[ 2(AF+BD+CE ) = \ell \]](http://latex.artofproblemsolving.com/9/7/4/974ff5c3a9064b29f3b47499d21f630e2c268bae.png)
if and only if is collinear with the incenter and the circumcenter of the triangle .
can be represented as 
for some real 
(we can regard this as a relation between the coordinates of 
on the axis 
.
(this relation takes place on 
,
)
;
)
for 
,
because of 
satisfying the condition, where 
(maybe 
doesn't intersect 
does, so we assume WLOG it's the first one). We have 
and 
,
,
.
