and ![$e\in [-2,2]$](http://latex.artofproblemsolving.com/b/b/7/bb78c9dedb18af5e6a904f5492917921e41a7ebc.png)
such that 
Find the value of 
be a positive integer such that 
,![\[
A = \{a_1, a_2, a_3, \ldots, a_m\}
\]](http://latex.artofproblemsolving.com/1/7/9/179687a134d5495ffff0b44fe763a2bb8b3ddd9c.png)
consist of distinct positive integers not exceeding 2025. Suppose that for every 
,
,
,
as well. Prove that![\[
\frac{a_1 + a_2 + a_3 + \cdots + a_m}{m} \geq 1013.
\]](http://latex.artofproblemsolving.com/c/0/d/c0d190f78d94deac1b947ba57a2de51d0b06f0bd.png)
for which I can find nonnegative integers 
such that for all indices 
,
divides 
?
...")
integers - take the set of all integers with a digit sum of 
in base 
to the list.
be a set with a binary operation 
such that
for all 
.
for all 
![\[\sum_{m=1}^\infty\sum_{n=1}^\infty\dfrac{m^2n}{3^m(n3^m+m3^n)}.\]](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 
having the following properties:
the entries of 
there exists a 
and the determinant of 
Y by
Given a convex quadrilateral 
, in which 
. Let 
be the intersection point of the bisector of the external angle at the vertex 
with the line 
, and let 
be the intersection point of the bisector of the external angle at the vertex 
with the line 
. Prove that 
.
, in which 
. Let 
be the intersection point of the bisector of the external angle at the vertex 
with the line 
, and let 
be the intersection point of the bisector of the external angle at the vertex 
with the line 
. Prove that 
.
