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.![$P(x) \in \mathbb{Z}[x]$](http://latex.artofproblemsolving.com/3/b/4/3b4dadc0cf4b02930a03c0ff0ce95e8b7a740ebb.png)
be a polynomial. Suppose that 
for infinitely many pairs 
. Prove that the graph of function 
has a center of symmetry.
functions such that
for all real numbers 
be an integer. In a configuration of an 
board, each of the 
cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate 
counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of 
, the maximum number of good cells over all possible starting configurations.
board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to 
, and the sum of the numbers in each column is equal to . Define 
to be the largest value in row 
, and let 
. Similarly, define 
to be the largest value in column , and let 
.
?
are natural numbers such that 
and 
is odd.
is divisible by 
is divisible by 
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 
denote the distance between and the closest integer. Let 
and let 
. Show that there exist infinitely many pairs 
of indices such that 
and ![\[ \|x_n-x_m \|< \min \left( \varepsilon , \frac{1}{2|n-m|} \right).\]](http://latex.artofproblemsolving.com/f/7/6/f763811ae2936fc093be3a5a5bac532429411621.png)
such that: 
. Find the maximum: 
+1 w
such that 
and 
are all perfect square.
be an acute triangle. Points 
, and 
lie on a line in this order and satisfy 
. Let 
and 
be the midpoints of 
and 
, respectively. Suppose triangle 
is acute, and let 
be its orthocentre. Points 
and 
lie on lines 
and 
, respectively, such that 
and are concyclic and pairwise different, and 
and are concyclic and pairwise different. Prove that 
and are concyclic.
be reals such that 
and 
. Prove that
, let 
be all the positive integers smaller than that are coprime to . Find all 
such that
for all 
.
and 
.
be the sequence of real numbers defined by 
and
Determine whether there exists a positive integer 
for all 
.
converges. Then 
as 
.![\[a_1 + \cdots + a_m \geq a_1^2 + \cdots + a_m^2 = \frac{1}{a_{m+1}} \to \infty\]](http://latex.artofproblemsolving.com/b/2/6/b262541c7b14a144d4cd54d6294e673aaab01c69.png)
gives contradiction!
,