AoPS Academy
be circumcenter, orthocenter and A-HM point of triangle 
. Let 
intersect 
at 
, respectively. Let 
be the triangle formed by 
. Prove that 
is tangent to 
are given. Prove that one can only use straightedge (which draws a straight line passing two drawn points) and compass (which draws a circle with a chosen radius equal to the distance of two drawn points and centered at a drawn points) to construct a line and two points on it with length 
in only 
steps (Namely, the total number of circles and straight lines drawn is at most .)
that meet the following conditions:
appears exactly once in each octal digit of
be an acute triangle, and let its circumcircle be 
. On side 
, pick a point 
(distinct from 
and 
). The lines through tangent to (other than 
, if 
lies inside the angle at ) touch again at points 
and 
. Let 
meet 
at 
, and let 
meet 
at 
. Prove that the three lines 
, 
, and 
are concurrent.
for all real numbers 
, hence find the value of 
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 fulfills![\[
\frac{1}{m^2+3m+2}+\frac{1}{m^2+5m+6}+\frac{1}{m^2+7m+12}+\dots +\frac{1}{m^2+15m+56}+\frac{1}{m^2+17m+72} = \frac{8}{33}
\]](http://latex.artofproblemsolving.com/8/c/4/8c4821c428f4c98ce4e66b26168313739188cc03.png)
Hence find the value of .
where 
is a monic polynomial of degree 
satisfying 
where is a polynomial of smallest degree possible satisfying 
and 
, if polynomial satisfies 
where is a polynomial of smallest degree satisfying 
and 
if polynomial satisfies 
is a polynomial of degree 
such that 
Evaluate 
and 
.
and 
.
are are the lengths of the sides of a 
g
then![$$(n-2)\left[\sum_{i=1}^{n}{\frac{a_i^2}{(1-a_i)^2}}-\frac n{(n-1)^2}\right]\geq(2n-1)\left(\sum_{i=1}^{n}{\frac{a_i}{1-a_i}}-\frac n{n-1}\right)^2.$$](http://latex.artofproblemsolving.com/9/3/e/93e5529518807b92cd9e0382f5fb07a293128b04.png)
