[/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 
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 
and 
intersect at points 
.
and 
.
,
,
attaining all positive values and positive numbers 
are given. Numbers 
satisfy
for every 
.
.
?
Y by
For each three vertices of the 
-gon (where 
), the center of the circle passing through them is marked.
1. Any center coincides with one of two given points,
and 
. Prove that the original -gon is cyclic.
2. It is known that any center coincides with one of six given points. Prove that the original-gon is cyclic.
3. For each natural, give an example of an -gon with exactly 
different centers.
4. Is there a natural, such that some -gon has more than one but fewer than different centers?
-gon (where 
), the center of the circle passing through them is marked.1. Any center coincides with one of two given points,
and 
. Prove that the original -gon is cyclic.2. It is known that any center coincides with one of six given points. Prove that the original
-gon is cyclic.3. For each natural
, give an example of an -gon with exactly 
different centers.4. Is there a natural
, such that some -gon has more than one but fewer than different centers?
