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 a polygon formed by the edges of an infinite chessboard, which does not intersect itself. Let the numbers 
represent the number of unit squares that have exactly 
edges on the boundary of respectively. Find the largest real number 
such that the inequality 
holds for each polygon constructed with these conditions.
such that, for all 
,
, the function 
is defined as, 
and for all 
, if the prime factorization of is 
then 
. We have an arithmetic sequence 
. If for a positive integer 
, the sequence 
is also an arithmetic sequence, show that the sequence 
has to be constant.
, 
are coprime positive integers. Find all 
such that:
such that![\[f(f(f(\frac{x+y}{2}))+x+y)=f(x)+f(y)+f(\frac{x+y}{2})\]](http://latex.artofproblemsolving.com/6/2/4/624afa78b68f810aaaecf0e1e2251047beeddc36.png)
for all rational numbers 
.
vertices, each edge has one of the colors 
, 
, or 
. For each 
, if the vertices can be divided into 
groups such that any two vertices connected by an edge of color 
are in different groups, find the minimum possible value of 
.
+1 w
be given pairwise coprime positive integers where 
. Let 
be positive integers. We call 
to be a grandson of if and only if, for all possible piles of stones whose total mass adds up to and consist of stones with masses , it's possible to take some of the stones out from this pile in a way that in the end, we can obtain a new pile of stones with total mass of . Find the greatest possible number that doesn't have any grandsons.
, for which 
for all functions 
, who satisfy the equation 
for all non-negative real numbers 
and 
.
and 
be sequences of real numbers for which 
and
for all positive integers . Prove that is eventually increasing (that is, there exists a positive integer 
for which 
for all 
).
and 
be two concentric circles. Let 
and 
be any two equilateral triangles inscribed in and respectively. If and 
are any two points on and respectively, show that ![\[ P'A^2 + P'B^2 + P'C^2 = A'P^2 + B'P^2 + C'P^2. \]](http://latex.artofproblemsolving.com/2/a/7/2a73de753f0332329b743d494272d3fb85f60ef9.png)
(meaning 
takes pairs of positive real numbers to positive real numbers) such that for any real numbers 
,
holds; and
then 
.
, 
, 
, 
and 
. Extend 
to meet the circumcircle of triangle 
at 
. Prove that 
.
be a set of at least three points of the plane in general position. Prove that there exists a non-intersecting polygon whose vertices are exactly the points of .
inside the convex hull not collinear with any two points in , then shift everything so that goes to the origin. Then sort the points in by argument and connect them that way.
in term of a function 
such that 
Prove that
