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 
?
.
, which implies that 
(since 
), or c=0,9.
since 
.
work.
, which is a contradiction since 
.
.
whereas 
(this occurs when a,k are both minimal at 1), whence there are no solutions for this case.
, i.e. m|n if n is a multiple of 1000.
and 
.
and 
.
Is there any generalizations?
be the four digit number formed from 
is an integer.
.
obviously has the same first two digits as 
.
.
,
,
.
,
,
.
gives 
,
.