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.
by 
, 
for every positive integer 
. Prove that 
leave distinct remainders when divided by 
.
+1 w
of distinct positive integers such that there exists a positive integer 
for which the number of divisors of 
and of 
are equal.
and 
be two sequences of integers such that 
and 
for all 
. Suppose that 
divides 
for an infinity of integers . Prove that there exist an integer 
such that 
for all
are coprime. Consider the sequence 
given by 
. Suppose that there exist positive integers 
such that 
is even and 
is an integer. Prove that the numerator in its simplest form of 
is an odd integer greater than 
.
be an integer and let 
be distinct subsets of 
. Show that there exists 
such that the n subsets 
are also disjoint.
satisfies 
. If 
, find the maximum possible value of 
.
such that for all irrational numbers 
and 
,
compute the smallest possible value of ![\[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\]](http://latex.artofproblemsolving.com/5/6/2/562c27c7779285b6439c4087f52933a41f29ee2e.png)
over all permutations 
of 
with incenter 
, the incircle 
touches sides 
and 
at points 
and 
, respectively. A circle passing through 
and 
touches at point 
. The circumcircle of 
meets 
at 
. Prove that 
is parallel to 
.
be a convex polygon, 
, 
, 
, 
, 
. Denote by: 
, 
, 
, 
, 
, . Prove what if
then a circle can be circumscribed around this polygon.
be a convex polygon, , , , , . Denote by: , , , , , . Prove what ifthen a circle can be circumscribed around this polygon.
and 
.
such that ![\[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]](http://latex.artofproblemsolving.com/0/6/2/0623b7c328af6d088cfc5dd288afe3e17a94ff29.png)
for all reals 
and 
.
s
.
where 
satisfy that 
.![$$\sum_{cyc} \sqrt[4]{\frac{a^3}{b+c}}$$](http://latex.artofproblemsolving.com/e/f/0/ef089c4f2915bbf75b0f60b044bbe9273b301a81.png)
