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the point 
is the centre of the excircle opposite the vertex 
This excircle is tangent to the side 
at 
, and to the lines 
and 
at 
and 
, respectively. The lines 
and 
meet at 
, and the lines 
and 
meet at 
Let 
be the point of intersection of the lines 
and , and let 
be the point of intersection of the lines 
and 
Prove that is the midpoint of 
opposite the vertex 
is the circle that is tangent to the line segment , to the ray beyond 
, and to the ray beyond 
.)
be complex numbers with 
. Find the greatest possible value of the expression 
digit numbers are there
? and 
?
and 
then is this uniformly convergence on 
comment on uniformly convergence on ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
where in general it is should be uniformly convergence.
and trying to solve
integer matrices having the following properties:
the entries of are (positive) prime numbers,
there exists a integer matrix such that 
and the determinant of is the square of a prime number.
such that the equation in the unknown 
:
admits 
roots: 
s.t. 
.
be the Fermat point of a 
. Prove that the Euler line of the triangles 
, 
, 
are concurrent and the point of concurrence is 
, the centroid of .
until 
such that 
is equilateral now, taking as homothetic center the midpoint of (3:1) sending X to circumcenter of , P to centroid of and A to centroid of that's enough
, 
and 
are the bisectors of 
, 
cutting 
, at 
, 
respectively. If 
and 
, find the angles of 
, and are the bisectors of , cutting , at , respectively. If and , find the angles of
, be the midpoint of . Prove that
.
and 
are points on the circumcircle of such that 
is parallel to . Prove that 
is perpendicular to the Simson line of .
be the Fermat point of a . Prove that the Euler line of the triangles , , are concurrent and the point of concurrence is , the centroid of . then the problem is trivial by Liang-Zelich , consider the midpoint of , 
, as 
,let the centroid of be 
then we know 
so 
is parallel to 
which is also parallel to 
. now meets the point outside of , as 
such that 
is an equilateral triangle, and now by 
, where 
is the midpoint of we have 
is parallel to so 
lies on therefore it's done
be an acute triangle whose incircle touches sides and at and , respectively. Let the angle bisectors of 
and 
meet 
at and 
, respectively, and let the midpoint of be 
. Show that 
is equilateral if and only if 
.
are drawn perpendicular to the sides 
respectively of then prove that
Not been able to solve this even though I know I can do this.
is parallel to , show that tan tan 
.
but nvm
be the orthocenter of and let 
be the circumcenter
from and be and 
then just use 
, is the circumcentre and is the orthocentre. Then, prove that 
.
be the antipode and then 
is a rhombus
is parallel to , show that tan tan . but nvm be the orthocenter of and let be the circumcenter from and be and then just use 
![\[\sum_{m=1}^\infty\sum_{n=1}^\infty\dfrac{m^2n}{3^m(n3^m+m3^n)}.\]](http://latex.artofproblemsolving.com/7/d/6/7d6247fbcfd6eac156ce774f727d7e95c470de52.png)
![$\underset{n\to \infty }{\mathop{\lim }}\,\sqrt[n]{lcm(1,2,3,...,n)}=e$](http://latex.artofproblemsolving.com/a/1/3/a139e5cb990f3386de109c701f8edc71b4054712.png)
now they are the circumcenters of the equilateral triangle formed outside of 
,
,
is concurrent at 
.
,