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.
points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least 
scalene triangles with vertices of that color.
and 
.
?
be positive integers satisfying ![\[\frac{ab}{a+b}+\frac{cd}{c+d}=\frac{(a+b)(c+d)}{a+b+c+d}.\]](http://latex.artofproblemsolving.com/b/8/c/b8c3fe14640a2dcb30b755b2347721da5d080a78.png)
Determine all possible values of 
.
+1 w
users, some pairs of whom are friends. Whenever user 
is friends with user 
, user is also friends with user . Events of the following kind may happen repeatedly, one at a time:, , and 
such that is friends with both and , but and are not friends, change their friendship statuses such that and are now friends, but is no longer friends with , and no longer friends with . All other friendship statuses are unchanged.
users have 
friends each, and users have friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.
and natural 
such that 
and 
, then find the minimum value of ![$\sqrt{ad^3 g^4} +\sqrt[3]{be^3 h^4} + \sqrt[8]{cf^3 k^4}$](http://latex.artofproblemsolving.com/b/f/a/bfa17e8bb4cc05a84ed6a757f506d6d0ed2b0081.png)
, satisfying the condition
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.
such that for all any 
be non-negative real numbers such that 
Prove that
satisfying![\[1=\frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb\]](http://latex.artofproblemsolving.com/a/e/6/ae65e3d53931fe9d60d0de76dc79540ccf44c26e.png)
and let 
denote the largest real number satisfying 
for all positive integers 
. What are the possible values of across all possible choices of the sequence 
?
and 
.
and 
.
Is there any generalizations?
?
?
?
where 
divisible by 
prime numbers prime
is a prime number, show that 
is composite.
or 
.
,
,
which is impossible, so 
.
.
such that 
is a perfect cube.
,
,
.
where p is a prime,then 
where 
we have 
,
If for an odd prime 
,
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 
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
.