[/quote]
,
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 
such that the inequality 
holds for each polygon constructed with these conditions.
such that, for all 
,
is defined as, 
and for all 
,
then 
.
.
,
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 
.
in term of a function 
such that 
vertices, each edge has one of the colors 
,
,
.
,
groups such that any two vertices connected by an edge of color 
are in different groups, find the minimum possible value of 
.
be given pairwise coprime positive integers where 
.
be positive integers. We call 
to be a grandson of 
for all functions 
,
for all non-negative real numbers 
and 
.
and 
be sequences of real numbers for which 
and
for all positive integers 
for which 
for all 
)
and 
be two concentric circles. Let 
and 
be any two equilateral triangles inscribed in 
are any two points on ![\[ P'A^2 + P'B^2 + P'C^2 = A'P^2 + B'P^2 + C'P^2. \]](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 
.
Prove that
,
,
,
and 
.
to meet the circumcircle of triangle 
at 
.
.
Y by
There are 
people at a round table. Some of them are knights who always tell the truth, and the rest are liars who always tell lies. Each of those sitting said the phrase: “among the 
people sitting clockwise from where I sit there are as many knights as among the people seated counterclockwise from where I sit”. For what 
could this happen?
people at a round table. Some of them are knights who always tell the truth, and the rest are liars who always tell lies. Each of those sitting said the phrase: “among the 
people sitting clockwise from where I sit there are as many knights as among the people seated counterclockwise from where I sit”. For what 
could this happen?
