[/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 
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
Given an acute triangle 
with 
as its circumcenter. Line 
intersects 
at 
. Points 
, 
are on 
, 
respectively such that 
, , , are concyclic. Prove that the length of the projection of line segment 
on side does not depend on the positions of and .
with 
as its circumcenter. Line 
intersects 
at 
. Points 
, 
are on 
, 
respectively such that 
, , , are concyclic. Prove that the length of the projection of line segment 
on side does not depend on the positions of and .
and 
be the projections of 
in 
and 
.
is cyclic, 
,
,
and 
are similiar. If we proved that the projections of 
and 
have the same size, we would know that the projections of 
and 
have the same size, which suffices. Now let 
and 
.
.
,
.
.![\[XY=EF \sin{\angle{FEX}} = EF \sin(\angle{FED}+\angle{DEX})\]](http://latex.artofproblemsolving.com/b/5/a/b5a615034961e4dcbb35060ed3a5b9182579dbdd.png)
![\[ = EF \left(\frac{BX \cdot EX + EX \cdot DX}{BE \cdot DE}\right)=DB\cos{\angle{BEX}} \cdot \frac{EF}{ED}\]](http://latex.artofproblemsolving.com/4/c/b/4cb534098fd6b74b979ce17401650b75614f1b11.png)
![\[= DB\cos{\angle{BEX}} \cdot \frac{\sin(180-\angle A)}{\sin \angle DAB}\]](http://latex.artofproblemsolving.com/b/b/2/bb21af840fe92256d3153f2b1d7de624f08774f9.png)
.
are constant, so 
must as well be constant.
:
and 
.
and 
and 
.
and 
.
,
and 
.
Note that by Power of a Point, 
,
.
.
Using what we derived at the beginning,
It follows that 
is independent of 
is also independent of 
.
.
on 
on 
and 
are cyclic quadrilaterals, we have 
.
.
is the intersection of 
.
,
as cyclic. Similarly, 
is also cyclic.
(as 
(as 
.
be the circumradius 
be the circumradius of 
)
,
)
.
.
and 
.
.
.
.
.
is equal for two arbitrary pairs of points 
If the problem would have changed from using circumcenter 
,
.
intersects 
for the second time again at 
.
is constant.
are isogonal, 
.![\[
XY = \frac{XY}{PD}\cdot PD = \frac{EF}{QD}\cdot PD = \frac{\sin A}{\sin\angle PAD}\cdot PD,
\]](http://latex.artofproblemsolving.com/9/2/4/924a49f97e25d7b00af729b6a5f650d46a191198.png)
which is indeed constant. Yay.
