be a triangle with orthocenter 
and let 
be the midpoint of 
Suppose that 
and 
are distinct points on the circle with diameter 
different from 
such that 
Prove that the orthocenter of 
lies on the circumcircle of 
.
.
Where 
![$f(x)\in R[x]$](http://latex.artofproblemsolving.com/a/1/5/a15dc97c051e81ae9ff0c060eaa4367bc723ccc9.png)
show that 
has at least one root of multiplicity one
is never a square of an integer.
for an infinite number of 
:
such that 
is a scalar
matrices
matrices![$\{x_n\} \subseteq [0,1]$](http://latex.artofproblemsolving.com/8/e/4/8e4206325387485d73765c9d7070ecb1ce18ff60.png)
,
,
,
and 
could be chosen to satisfy 
and 
.
are complex numbers with absolute value 
.
define 
.
converges. Show that,
.
![$g_i \in C([0,1],\mathbb R_+)$](http://latex.artofproblemsolving.com/3/7/9/379b9e3f56dd862fa6b403bf9b3f799ffc109b2d.png)
convex.![$a \in [0;1]$](http://latex.artofproblemsolving.com/a/f/b/afbcf487b151bb65405d3d385abf89b9393ce9af.png)
with 
?
........
?
Y by kiyoras_2001
Kuba has two finite families 
of convex polygons (in the plane). It turns out that every point of the plane lies in the same number of elements of 
as elements of 
. Prove that 
.
\textit{Note:} We treat segments and points as degenerate convex polygons, and they can be elements of or .
of convex polygons (in the plane). It turns out that every point of the plane lies in the same number of elements of 
as elements of 
. Prove that 
.\textit{Note:} We treat segments and points as degenerate convex polygons, and they can be elements of
or .
