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.
of positive integers is called central if for every positive integer 
, the arithmetic mean of the first 
terms of the sequence is equal to .
of positive integers such that for every central sequence 
there are infinitely many positive integers with 
.
be a set of 
points in the plane, no three of which are collinear. Initially these points are connected with segments so that each point in is the endpoint of exactly two segments. Then, at each step, one may choose two segments 
and 
sharing a common interior point and replace them by the segments 
and 
if none of them is present at this moment. Prove that it is impossible to perform 
or more such moves.
is the center of the inscribed circle and holds 
. Prove that 
.
be a primes ,
be positive integers such that 
and 
.
be an acute triangle with incentre 
and 
. Let lines 
and 
intersect the circumcircle of at 
and 
, respectively. Consider points 
and 
such that 
and 
are parallelograms (with 
, and 
). Let 
be the point of intersection of lines 
and 
. Prove that points 
, and are concyclic.
, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
is the largest positive integer that divides both 
and 
. Integers and are coprime if 
., 
, 
, and 
is the point such that 
and 
, where and 
lie on the same side of . Let 
be the point on such that 
, and let be the midpoint of 
. Prove that the line through parallel to 
passes through the midpoint of .
.
, there exist indices 
such that 
is divisible by .
, find the minimum possible value of 
.
be an acute triangle with the 
mixtilinear incircles are 
respectively. 
is tangent to the circumcircle of at 
. 
are the centers of circles 
respectively. Suppose the reflection of line 
over 
intersects the reflection of line 
over 
at point 
Prove that: 
, no two of which are 
. Prove that:
and 
such that
be a sequence of positive real numbers such that for every 
, we have:![\[
a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i}
\]](http://latex.artofproblemsolving.com/e/3/0/e30e9e533eb9a41cf847484e676ac4522edda665.png)
Prove that there exists a natural number 
such that for all 
, the following holds:![\[
a_n < \frac{1}{1404}
\]](http://latex.artofproblemsolving.com/0/1/f/01f1a81eed2230332d51a15501ef2a6ad3a7ec82.png)
such that 
Prove that
?
?
?
where 
divisible by 
prime numbers prime
is a prime number, show that 
is composite.
or 
.
,
,
which is impossible, so 
.
.
is a perfect cube.
,
,
.
where p is a prime,then 
where 
we have 
,
If for an odd prime 
we have 
,
which is also impossible(it is also clear that k=0 or k=1).Then 
.
then 
,
.
or 
It is trivial that 
.
.
has residue 
mod 
,
.
,
.
doesn't divide 
,
so 
,
.
take 
take 
take 
a succession of positive integers such that for every couple of positive integers 
we have 
. Prove that there exists a positive integer 
such that 
.
is obvious. Let 
be called a good pair if 
for some 
.
are 
distinct pairwise good integers.
is a good pair iff 
.
,
then let 
,
to both sides gives 
so the claim is proven.
,
for 
and 
.
are 
pairwise good integers. First we show that 
are pairwise good. This follows from 
by induction and our definition of 
.
is a good pair. Note that 
as desired, so we have 
.
and 
such that 
and 
we have![\[
\text{gcd}(x+y,mn)>1
\]](http://latex.artofproblemsolving.com/0/e/3/0e3bae9e964367fac00f7f728b1db4fecf514ba3.png)
be the subset of primes dividing 
the subset of primes dividing 
such the two are disjoint. Further, take 
to be the set of overlap in 
prime factors. If none exist take it to be 
.
hence 
,
.
such for any 
:
.
and 
are coprime and 
,
-
.
.
t
.
has a period of 
because it is a fermat prime
such that 
is clearly a solution.Let 
.
.
and then 
or simply 
,
,
the number 
has at least 
.