AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be the maximal number of the vertices of a convex lattice polygon with exactly 
lattice points in the interior. Show that:
for 
for some constant 
.
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 
, for 
in at least 
different ways.
be positive real numbers satisfying 
. Prove that![\[
3(x^{\frac{2}{3}} + y^{\frac{2}{3}}) \geq 4 + 2x^{\frac{1}{3}}y^{\frac{1}{3}}.
\]](http://latex.artofproblemsolving.com/f/3/c/f3cd114f909f388eacee142d3f90d07f6b149131.png)
be real numbers such that 
. Prove that
Let be real numbers such that 
. Prove that
and , we define 
as follows:
where 
are all the positive divisors of . For example,
.
Find all positive integers such that 
.
Find all positive integers such that 
.
be a convex quadrilateral. Suppose that the circles with diameters 
and 
intersect at points 
and 
. Let 
and 
. Prove that the points 
, 
, and are concyclic. and are not the diagnols)
attaining all positive values and positive numbers 
are given. Numbers 
satisfy
for every 
. Prove that there exists a positive integer satisfying 
.
?
and colors are given. We will say that a set of 
points on a plane is 
, if it contains exactly 2 points of each color and if lines connecting every two points of the same color are pairwise distinct. Find, in terms of the least integer such that: in every set of 
points of a plane, no three of which are collinear, consisting of points of every color there exists a subset.
be a triangle, 
the center of the excircle at side 
, and 
its reflection across . Prove that 
is parallel to the Euler line of the triangle 
.
by the Incenter-Excenter Lemma by using the Law of Sines. 
Here, 
. So, if 
is the orthocenter of 
, we have 
Also, 
It can be shown that these two ratios are equal and the conclusion follows.
be a triangle, the center of the excircle at side , and its reflection across . Prove that is parallel to the Euler line of the triangle .
and a reflection over the angle bisector of 
. Then the circumcenter 
of 
goes to and its orthocenter is mapped to 
. Since 
are isogonal, lies on 
and lies on 
. By the inversion, note that 
, so by Thales' theorem, we are done.
and 
denote 
and 
excenters of 
and denote incenter of , circumcenter and orthocenter of 
be foot of altitude from to is orthic triangle of 
and 
is orthocenter of and are corresponding points in similar triangles 
and , and so are and 
is cyclic with diameter 
is Euler line of so we are done
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 
.
