AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
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)
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.
such that there exist positive integers 
for which the following holds 
for all 
.
where 
is a perfect cube
of a prime number 
and a positive integer 
for which 
is an integer.
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 
.)
such that if 
and 
are distinct rational numbers satisfying 
or 
, then 
? Justify your answer.
be a positive integer. Consider the following equation: ![\[ \{x\}+\{2x\}+ \dots + \{nx\} = \lfloor x \rfloor + \lfloor 2x \rfloor + \dots + \lfloor 2nx \rfloor\]](http://latex.artofproblemsolving.com/8/2/9/8299b3ceb1650418c0a373536cb1d2931900efd9.png)
a) For 
, solve the given equation in 
., the equation has at most 
real solutions.
and 
intersect at two points 
and 
. Let 
be the line tangent to these circles at 
and 
, respectively, so that lies closer to than . Let 
be the line parallel to and passing through the point , with 
on and 
on . Lines 
and 
meet at 
; lines 
and meet at 
; lines 
and meet at 
. Show that 
.
. How many complex numbers 
are there such that 
and 
is an integer.
, 
, and 
be complex numbers such that they satisfy these equations:
If the maximum of 
can be expressed as 
for positive integers 
, 
, and 
, find the minimum possible value of 
.
, , and be complex numbers such that they satisfy these equations:
If the maximum of can be expressed as for positive integers , , and , find the minimum possible value of .
and 
and 
.
; multiplying: 
.
.
and 
.
substitution as @above, we get 
.
and let 
.
,
.
giving the answer of 
.