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 normed vector spaces, where 
is a Banach space, and let 
.![\[
X = \left\{ x \in E : \sup_{n \geq 1} \|A_n x\| < +\infty \right\}
\]](http://latex.artofproblemsolving.com/1/4/f/14f368e30ebb290aa9b803da77e96afe63b0c7c0.png)
is either an empty set or second category.
be an integer. Ivan writes the numbers 
each on different cards. He then shuffles these 
cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.
integers 
, each less than or equal to 
, prove that at least one of them is divisible by some other member of the set.
from ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
to 
that is pointwise bounded
must also be uniformly bounded 
If no, find a counterexample.
is tangent to the circle 
at 
. 
and 
cut the circle at 
and 
respectively. 
is the in-centre of 
, and is on the line 
. 
and 
intersect at 
, while 
and 
intersect at 
. Prove that the points 
lie on a circle.
be the number of positive divisors of 
. Let 
be the number of positive divisors of which have remainders 
when divided by 
. Find all positive integral values of the fraction 
.
kids in a class. Is it true that for all positive integers 
it is possible to create several clubs each with 3 kids such that any pair of kids are both present in exactly one club?
has a right angle at 
. The centre of the circumcircle is called 
, and the base point of the normal from to is called . The point lies on 
with 
. The angle bisector of 
meets 
in 
. The lines 
and 
intersect in . Show that the lines 
and 
are parallel.
let 
denote the number of (positive) divisors of 
. Find all functions 
with the following properties: [list][*] 
for all 
.
divides 
for all 
, 
.[/list] be a triangle with circumcenter and orthocenter 
. 
the altitudes of . A point 
lies on line 
such that 
. A point 
lies on the circumcircle of such that 
are concurrent. 
meets 
at 
. Prove that 
are concyclic.
be a triangle. Distinct points , , lie on sides , , and , respectively, such that quadrilaterals 
and 
are cyclic. Line 
meets the circumcircle of again at . Let denote the reflection of across . Show that lies on the circumcircle of 
.![\[
\frac{(2n)!}{n! \cdot (n+1)!}
\]](http://latex.artofproblemsolving.com/b/9/7/b974bd023a1bf175e6dd0561e4a3a79a55ff7098.png)
is always an integer. Prove that for every 
, there are infinitely many values of 
such that![\[
\frac{(2n)!}{n! \cdot (n+k)!}
\]](http://latex.artofproblemsolving.com/5/f/9/5f9fb4566dfc29e19f658402b4f0bb702441ee62.png)
is an integer.
