[/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
Circle 
(the center is .) is inside the 
and it is tangent to the two sides of the angle. Let 
be the circle that passes through the center of and tangent to two sides of angle and let 
be one of the endpoint of diameter of that passes through and 
be the intersection of this diameter and circle 
Prove that the cirlce that is the center and 
is the radius is also tangent to the two sides of 
(the center is .) is inside the 
and it is tangent to the two sides of the angle. Let 
be the circle that passes through the center of and tangent to two sides of angle and let 
be one of the endpoint of diameter of that passes through and 
be the intersection of this diameter and circle 
Prove that the cirlce that is the center and 
is the radius is also tangent to the two sides of 
