AoPS Academy
, 
, 
, 
, 
, 
be positive integers and let 
.
divides 
and 
. Prove that is composite.
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 positive reals such that 
. Find the minimum and maximum value of
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. 
.
![\[\sum_{m=1}^\infty\sum_{n=1}^\infty\dfrac{m^2n}{3^m(n3^m+m3^n)}.\]](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)
,such that 
Prove that : 
Where 
is a positive real number solution of equation : 
![\[
u=1+\sqrt{1+ka},\quad v=1+\sqrt{1+kb},\quad w=1+\sqrt{1+kc},
\]](http://latex.artofproblemsolving.com/1/3/e/13e016c14c1ba8467ac11bcd217e114fcd067a8d.png)
and note that![\[
a=\frac{(u-1)^2-1}{k}=\frac{u^2\Bigl(1-\frac 2u\Bigr)}{k}= \frac{u^2\Bigl(\frac{2}{v}+\frac{2}{w}\Bigr)}{k}.
\]](http://latex.artofproblemsolving.com/0/2/f/02f24b6ce57cca2486ab567b3b0f9aa7b4573902.png)
![\[
xyz=\sum ax^2=\frac{1}{k}\sum \Bigl(\frac{2}{v}+\frac{2}{w}\Bigr)(ux)^2=\frac{1}{k}\sum \frac{2}{u}\Bigl[(vy)^2+(wz)^2\Bigr]\ge \frac{1}{k}\sum\frac{2}{u}\cdot2vywz.
\]](http://latex.artofproblemsolving.com/3/d/2/3d22da4a6db42ca431539f792aafd26f7e1a2d9d.png)
yields![\[
1\ge\frac{uvw}{k}\sum \frac{4}{u^2 x}.
\]](http://latex.artofproblemsolving.com/7/8/6/786bfb004ad038ed6fe576141a753e380792ac3f.png)
![\[
\sum x\ge \frac{uvw}{k}\Biggl(\sum \frac{4}{u^2 x}\Biggr)\Biggl(\sum x\Biggr)\ge \frac{uvw}{k}\Biggl(\sum \frac{2}{u}\Biggr)^2=\frac{uvw}{k}.
\]](http://latex.artofproblemsolving.com/e/b/d/ebd9b73d8809fa523f11604c2e7ed47b742b01a8.png)