?
and 
Prove that![$$ a+b+kc^3\geq\sqrt[4]{\frac{4k} {27}}$$](http://latex.artofproblemsolving.com/5/6/1/561a990e5885863c77d454ffa3f1973e1bbf60b4.png)
![$$ a+b+kc^4\geq\frac{5} {8}\sqrt[5]{\frac{k} {2}}$$](http://latex.artofproblemsolving.com/4/b/1/4b14911672688250c909571d788aaf10724462eb.png)
Where 
be a trapezoid such that 
lie on a circle with center 
,
is parallel to side 
.
and 
intersect at point 
,
.
.
,
and 
are positive integers and 
.
Prove that
and height 
Another sphere of radius 
is internally tangent to the lateral surface of the cone, but does not intersect the larger inscribed sphere. A plane is tangent to both of these spheres, and passes through the inside of the cone. The intersection of the plane and the cone forms an ellipse. Find the maximum area of this ellipse.
Y by
Today I was browsing the forum High School Olympiads. There's a user whose username is the_universe6626, but then, when I flicked down to browse earlier answers today(note: I didn't 'search for this user's posts'), I found that one of them was labeled '23 posts'. I browsed his earlier answers today, and the number of posts increases, from 25 posts in the morning all the way to 36 posts right now. Is this a feature or a bug? If it is a feature, then certainly I haven't seen it elsewhere.
