AoPS Academy
for which I can find nonnegative integers 
such that for all indices 
, 
divides 
?
...") for which I there exists nonnegative integers such that for all indices , divides ?
integers - take the set of all integers with a digit sum of 
in base , then append 
to the list.
be a positive integer such that 
, and let the set![\[
A = \{a_1, a_2, a_3, \ldots, a_m\}
\]](http://latex.artofproblemsolving.com/1/7/9/179687a134d5495ffff0b44fe763a2bb8b3ddd9c.png)
consist of distinct positive integers not exceeding 2025. Suppose that for every 
, with 
, if 
, then 
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)
. How many complex numbers 
are there such that 
and 
is an integer.
be the midpoint of 
of the convex quadrilateral 
. It is known that the lines 
and 
are perpendicular and 
. Prove that the lines 
and 
are parallel.
