AoPS Academy
be an acute triangle. Define 
as its incenter. Let 
and 
be the incircle's tangent points to 
and 
, respectively. Let 
be the midpoint of 
. Let 
be the intersection point of a perpendicular line passing through to 
. Line 
intersects the circumcircle of 
at 
. The circumcircle of 
intersects line 
at 
. Prove that quadrilateral 
is cyclic.
for which there exists a sequence of distinct positive integers 
, 
, 
, 
satisfying ![\[s_1^{s_2}=s_2^{s_3}=\cdots=s_{n-1}^{s_n}.\]](http://latex.artofproblemsolving.com/a/f/5/af594ff07aa2f89cb70ae669263558049556cf0e.png)
be an integer and 
is a sequence of real numbers for which 
and:
Prove that all elements of the sequence are positive.
be a positive integer, and let 
be a subset of 
. An -partition of into 
parts is a representation of n as a sum 
, where the parts 
belong to and are not necessarily distinct. The number of different parts in such a partition is the number of (distinct) elements in the set 
.-partition of into parts is optimal if there is no -partition of into 
parts with 
. Prove that any optimal -partition of contains at most ![$\sqrt[3]{6n}$](http://latex.artofproblemsolving.com/2/0/1/201489b1f95ad200036698a606f712da7bb465b9.png)
different parts.
