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 an integer, and let 
be positive real numbers such that 
. Prove that![\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]](http://latex.artofproblemsolving.com/7/7/f/77fb719e975225b59c64f1f3aa102c98a6c25f50.png)
not be a trapezoid such that there is a circle centered at 
that is tangent to the four sides 
, 
, 
, and 
. Let 
, 
, 
, and 
be the circumcenters of the triangles 
, 
, 
, and 
, respectively. Prove that there is a circle containing the circumcenters of the triangles 
, 
, 
, and 
.
such that
satisfies 
. Find the value of 
.
with 
, a tangent to the circumcircle of at point 
intersects the extension of 
at point 
. 
is the midpoint of 
, and 
intersects the circumcircle of at 
. Prove that 
.
be an acute triangle inscribed in a circle 
. Let 
be the orthogonal projection of 
on 
. Let 
be the orthogonal projection of on 
respectively. Let be a point such that 
is the diameter of . Prove that the area of the quadrilateral 
is equal to the area of triangle .
be the number of ways in which the set 
can be partitioned into non-empty subsets. Let 
be the number of ways in which the set 
can be partitioned into non-empty subsets such that consecutive numbers belong to distinct subsets. Partitions that differ only in the order of the subsets are considered equal. Prove that 
.
right-angled at 
, the median through bisects the angle between 
and the bisector of 
. Prove that 
.
and 
Prove that
Where 
)
and 
Prove that
![$$a+\dfrac{2}{a}+\dfrac{3}{b}\geq 4\sqrt[4]{\frac23}$$](http://latex.artofproblemsolving.com/b/c/1/bc1ceb6a41d5fc0598cdd0cc455875087f9d4585.png)
![$$\dfrac{2}{a}+\dfrac{3}{b}\geq 2\sqrt[4]3$$](http://latex.artofproblemsolving.com/f/2/2/f22009d039177d6343c55023754310e0e59c1744.png)
![$$3a+\dfrac{2}{a}+\dfrac{3}{b}\geq \sqrt[4]{354+66\sqrt{33}}$$](http://latex.artofproblemsolving.com/e/c/9/ec9133ad1679ba58cefdb5de88de5550e6903d83.png)
such that 
.
Where 
