[/quote]
,
,
be the orthocenter of 
.
.
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
A triangle 
is given. A circle 
passes through 
and is tangent to the side 
at a point 
. The circle intersects 
and 
at points 
and 
respectively. Prove that the (minor) arcs 
and 
are equal if and only if is tangent to the circumcircle of at .
is given. A circle 
passes through 
and is tangent to the side 
at a point 
. The circle intersects 
and 
at points 
and 
respectively. Prove that the (minor) arcs 
and 
are equal if and only if is tangent to the circumcircle of at .
