[/quote]
,
be circumscribed about circle 
,
be the orthocenter of 
.
at 
.
at 
.
be the midpoint of 
,
meet 
.
at 
.
is tangent to 
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 the maximal number of the vertices of a convex lattice polygon with exactly 
lattice points in the interior. Show that:
for 
.
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)
be reals such that 
and 
.
and a set 
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 real numbers such that 
.
Let 
.
and 
as follows:
where 
are all the positive divisors of 
.
Find all positive integers 
.
Find all positive integers 
.
and 
intersect at points 
.
and 
.
,
,
attaining all positive values and positive numbers 
are given. Numbers 
satisfy
for every 
.
.
?
Y by Love_Math1994, Adventure10, Mango247
Let 
be a cyclic quadrilateral with circumcentre 
and the pair of opposite sides not parallel with each other. Let 
and 
Denote, by 
the intersection of the internal angle bisectors of 
and 
and 
and 
and 
Suppose that the four points 
are distinct.
(a) Show that the four points are concyclic. Find the centre of this circle, and denote it as 
(b) Let
Prove that 
are collinear.
be a cyclic quadrilateral with circumcentre 
and the pair of opposite sides not parallel with each other. Let 
and 
Denote, by 
the intersection of the internal angle bisectors of 
and 
and 
and 
and 
Suppose that the four points 
are distinct.(a) Show that the four points
are concyclic. Find the centre of this circle, and denote it as 
(b) Let
Prove that 
are collinear.This post has been edited 2 times. Last edited by Potla, Feb 9, 2012, 6:20 AM
therefore 
are cyclic. This gives 
So, 
lies on the radical axis of 
and 
Similarly, 
lies on the radical axis, and therefore 
is the radical axis of these two circles, ie 
wrt 
and therefore 
Together, these give 
.
are midpoints of 
.
and 
are similar. And 
and 
are similar. So the configuration 
is similar to 
.
on 
intersects that of 
on 
at 
collinear points. Now note that, 
is homothetic to the triangle formed by the midpoints of the arcs 
.
intersects at the circumcenter of it. So done.
be 
respectively.
is excenter of 
and we obtain similar results for 
Then 
implying the cyclicity of 
Then since 
and 
are collinear with 
In order to show that 
as well, we prove a stronger result by showing that 
so 
is cyclic; thus by considering the radical center of 
we get 
done.
Thus, 
is cyclic. From the definition itself, it follows 
and 
Hence, 
is cyclic 
and 
and similarly, 
