[/quote]
,
,
.
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 
.
be a convex quadrilateral. Suppose that the circles with diameters 
intersect at points 
.
and 
.
,
,
attaining all positive values and positive numbers 
are given. Numbers 
satisfy
for every 
.
.
?
Y by Adventure10, Mango247
Let 
be the circumcircle of non-isosceles acute triangle 
. Tangent lines to in 
and 
intersect at point 
. Let M be the midpoint of 
, and 
be the orthocenter of triangle . The line 
intersects lines 
and 
at points 
and 
, respectively. The points 
and 
are defined analogously. Prove that 
and 
are the altitudes of triangle 
.
be the circumcircle of non-isosceles acute triangle 
. Tangent lines to in 
and 
intersect at point 
. Let M be the midpoint of 
, and 
be the orthocenter of triangle . The line 
intersects lines 
and 
at points 
and 
, respectively. The points 
and 
are defined analogously. Prove that 
and 
are the altitudes of triangle 
.This post has been edited 1 time. Last edited by sebepkaly, Jan 21, 2015, 10:11 AM
.
is the isogonal conjugate of 
WRT 
,
,
are concyclic .
,
is coaxial with 
and the degenerate circle 
,
of 
is the diameter of 
.
WRT 
and 
be the feet of the altitudes on 
is the antipode of 
on 
at 
Since 
But from parallelogram 
with center 
we obtain 
and similarly 
onto 
,
.
at 
,
,
.
Now lets draw, 
And 
,
is symmedian.Now 
must be collinear( if 
,
and 
,
be a triangle with 
as orthic triangle and orthocenter 
be midpoint of 
.
be foot from 
to 
.
be orthocenter WRT 
.
is 
s
Thus, 
Thus, 
