[/quote]
,
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 
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 
.
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 
Y by
Given a quadrilateral 
, around which you can circumscribe a circle. The perpendicular bisectors of sides 
and 
intersect at point 
and intersect sides 
and 
at points 
and 
resepctively. It turned out that the points 
lie on the same circle. Prove that the points 
lie on one line.
(Olena Artemchuk)
, around which you can circumscribe a circle. The perpendicular bisectors of sides 
and 
intersect at point 
and intersect sides 
and 
at points 
and 
resepctively. It turned out that the points 
lie on the same circle. Prove that the points 
lie on one line.(Olena Artemchuk)
This post has been edited 1 time. Last edited by parmenides51, Apr 1, 2021, 9:56 AM
