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where 
is a parallelogram.
.
is connected to the midpoint of ![$[CT]$](http://latex.artofproblemsolving.com/f/3/f/f3f8e9913dc696e543e6a34d169ec3deb276c84d.png)
, point 
to the midpoint of ![$[DT]$](http://latex.artofproblemsolving.com/6/d/d/6dd49550ab42ec2f047eaf4b357ac4e7d1387b1e.png)
,
to the midpoint of ![$[AT]$](http://latex.artofproblemsolving.com/6/e/3/6e397da3c2a38b457f7b3504c5cb2a488a64f3a6.png)
and point 
to the midpoint of ![$[BT]$](http://latex.artofproblemsolving.com/6/a/4/6a4b1bcea7921e762a359321ab861b6937f04782.png)
.
.
.
be a triangle of orthocenter 
. The circle of diameter 
and the circumcircle of triangle 
intersect a second time in 
.
passes through the midpoint of segment 
.
. Prove that
Let 
. Prove that
Let 
. Prove that
Let . Prove that
with edges 
. three perpendiculars to the three lateral faces. from the apex such that these three points lie above the apex.
and radius 
.
. and radius ..
exist that satisfy 
for all real 
and the domain of f is strictly within the set of 
, the range being 
?
with edge 
cm.
and tangent to the plane 
.
) stones. On their turn, each player must divide a pile. The player who can make all piles have at most 2 stones wins. Depending on n, determine which player has a winning strategy.
and 
be real numbers sach that 
and 
if 
where 
and 
are relatively prime positive integers, what is 
and 
defined, and with another symmetric equation in 
would be to attempt to write the third equation in terms of the symmetric sums of 
then substitute the first and third sums so we can find the second, its then pretty trivial to find 
.
