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 a triangle with circumcircle 
. Arbitrary points 
on 
respectively. Circumcircle 
of triangle 
intersects at 
. 
intersects 
at 
. 
cuts and at 
respectively. Construct parallelogram 
. Prove that 
are collinear.
be the unit circle in the complex plane. Let 
be the map given by 
. We define 
and 
for 
. The smallest positive integer 
such that 
is called the period of 
. Determine the total number of points in 
of period 
.
)
, and be positive real numbers, such that 
. Prove the inequality![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
be a positive integer such that divides the sum![\[
1 + \sum_{i=1}^{n-1} i^{n-1}.
\]](http://latex.artofproblemsolving.com/e/2/d/e2daf7800da3f984cd761ed1fecf10083444db25.png)
Prove that is square-free.
,such that the inequality 
holds for all positive reals 
and for every positive real 
, there exist positive reals 
, such that the equality holds.
be a triangle with 
. The tangent to the circumcircle of 
at 
intersects 
at 
. The angle bisector of 
intersect at 
. Suppose that the perpendicular bisector of 
intersect 
at 
, respectively. Show that 
be an interior point of the isosceles triangle with 
. If
prove that 
.
be positive real numbers such that 
. Prove that 
be a cube with an edge of 
m. Determine the minimum value of the length the segment with the ends on 
and 
and forms an angle of 
with the plane of the face 
.
with 
.![\[
\frac{a^5}{b^2 + 2c^3} + \frac{2b^5}{3c + a^6} + \frac{c^7}{a + b^4} \geq 2.
\]](http://latex.artofproblemsolving.com/f/f/5/ff5043705025592aeefc55a764a8b7170e13ff4b.png)
be real numbers such that 
.
.
Where 
