[/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 
.
be a convex quadrilateral. Suppose that the circles with diameters 
and 
intersect at points 
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 Adventure10
The board features a triangle 
, its center of the circle circumscribed is the point 
, the midpoint of the side 
is the point 
, and also some point 
on side 
(see fig.). Master knowing that 
of this triangle is equal to the sharp angle 
has separately drawn an angle equal to . After this teacher wiped the board, leaving only the points 
and the angle . Is it possible with a compass and a ruler to construct the triangle ? Justify the answer.
(Grigory Filippovsky)
, its center of the circle circumscribed is the point 
, the midpoint of the side 
is the point 
, and also some point 
on side 
(see fig.). Master knowing that 
of this triangle is equal to the sharp angle 
has separately drawn an angle equal to . After this teacher wiped the board, leaving only the points 
and the angle . Is it possible with a compass and a ruler to construct the triangle ? Justify the answer.(Grigory Filippovsky)
This post has been edited 2 times. Last edited by parmenides51, Feb 24, 2024, 8:00 PM
intersect at 
at 
.
satisfies the exact same properties as 
:
.
