[/quote]
,
be reals such that 
and 
.
and a set 
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 positive real numbers satisfying 
.![\[
3(x^{\frac{2}{3}} + y^{\frac{2}{3}}) \geq 4 + 2x^{\frac{1}{3}}y^{\frac{1}{3}}.
\]](http://latex.artofproblemsolving.com/f/3/c/f3cd114f909f388eacee142d3f90d07f6b149131.png)
be real numbers such that 
.
Let 
.
and 
,
as follows:
where 
are all the positive divisors of 
.
Find all positive integers 
.
Find all positive integers 
.
be a convex quadrilateral. Suppose that the circles with diameters 
intersect at points 
and 
.
and 
.
,
,
attaining all positive values and positive numbers 
are given. Numbers 
satisfy
for every 
.
.
?
and 
points on a plane is 
,
points of a plane, no three of which are collinear, consisting of 
,
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)
.
Y by Adventure10, Mango247
Let the incircle of the triangle 
touch ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
at 
and 
be the incenter of the triangle. Let 
be midpoint of ![$[ID]$](//latex.artofproblemsolving.com/3/9/c/39c0124bdd40abf098bae12cd10f3b575581439f.png)
. Let the perpendicular from to 
meet 
and 
at 
and 
, respectively. Let the perpendicular from to meet and at 
and 
, respectively. Show that 
.
touch ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
at 
and 
be the incenter of the triangle. Let 
be midpoint of ![$[ID]$](http://latex.artofproblemsolving.com/3/9/c/39c0124bdd40abf098bae12cd10f3b575581439f.png)
. Let the perpendicular from to 
meet 
and 
at 
and 
, respectively. Let the perpendicular from to meet and at 
and 
, respectively. Show that 
.
from 
of 
through 
cross ratio 
is harmonic. 
is clearly D-midline of 
meeting 
at its midpoint 
hence 
or 
But by Thales theorem 
and 
.
meet 
,
is semiperimeter.
and 
.
.
.
,
to 
to 
and 
.
.
.
and 
.
and 
.
.
.
where 
.
.
so 
by opening the brackets the desired conclusion results.
be the touch points of the incircle with 
be the foot from 
be the foot from 
we get 
, 
and 
.
.
.
we have 
and because 
we have 
.
and 
.
.
.
.
.
