and 
n
.
,
.
.
that meet the following conditions:
be an acute triangle, and let its circumcircle be 
.
,
(distinct from 
and 
)
,
lies inside the angle at 
and 
.
meet 
at 
,
meet 
at 
.
,
,
are concurrent.
for all real numbers 
,
if 
.
different stamps each.
he can use is in the range 
such that ![$\lfloor \sqrt[3]{n}\rfloor\mid n$](http://latex.artofproblemsolving.com/5/6/2/562cedff73965e1dce784a360514b931e36158b1.png)
.
be a positive integer, such that ![\[
\frac{1}{m^2+3m+2}+\frac{1}{m^2+5m+6}+\frac{1}{m^2+7m+12}+\dots +\frac{1}{m^2+15m+56}+\frac{1}{m^2+17m+72} = \frac{8}{33}
\]](http://latex.artofproblemsolving.com/8/c/4/8c4821c428f4c98ce4e66b26168313739188cc03.png)
Hence find the value of 
.
Y by
Let 
be an acute triangle with circumcenter 
and orthocenter 
. Suppose that 
. Let 
be an arbitrary point on line segment 
. Let 
and 
be points on 
and 
respectively such that 
. Prove that the circumcenter of 
lies on the perpendicular bisector of 
.
be an acute triangle with circumcenter 
and orthocenter 
. Suppose that 
. Let 
be an arbitrary point on line segment 
. Let 
and 
be points on 
and 
respectively such that 
. Prove that the circumcenter of 
lies on the perpendicular bisector of 
.
