the point 
is the centre of the excircle opposite the vertex 
This excircle is tangent to the side 
at 
,
and 
at 
and 
,
and 
meet at 
,
and 
meet at 
Let 
be the point of intersection of the lines 
and 
be the point of intersection of the lines 
and 
Prove that 
is the circle that is tangent to the line segment 
,
.
![\[\sum_{m=1}^\infty\sum_{n=1}^\infty\dfrac{m^2n}{3^m(n3^m+m3^n)}.\]](http://latex.artofproblemsolving.com/7/d/6/7d6247fbcfd6eac156ce774f727d7e95c470de52.png)
be complex numbers with 
.
![$\underset{n\to \infty }{\mathop{\lim }}\,\sqrt[n]{lcm(1,2,3,...,n)}=e$](http://latex.artofproblemsolving.com/a/1/3/a139e5cb990f3386de109c701f8edc71b4054712.png)
digit numbers are there
?
?
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 solv
integer matrices 
the entries of 
there exists a 
and the determinant of 
such that the equation in the unknown 
:
admits 
roots: 
s.t. 
.
Y by
Let 
be a triangle with altitudes 
intersect at orthocenter 
. Let 
be an arbitrary point lies on 
. 
be the reflection point of 
on 
. 
intersects 
at 
. Prove that 
passes through the circumcenter of 
.
be a triangle with altitudes 
intersect at orthocenter 
. Let 
be an arbitrary point lies on 
. 
be the reflection point of 
on 
. 
intersects 
at 
. Prove that 
passes through the circumcenter of 
.
This post has been edited 1 time. Last edited by ken3k06, May 25, 2022, 1:38 AM
,
and 
are symmetric wrt 
,
is tangent to 
.
and 
.
to 
,
and 
lie on 
and 
respectively. Let 
be the reflection of 
on 
,
are concyclic. From here, proving 
is easy.
