[/quote]
,
such that for all 
the following equality holds ![\[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \]](http://latex.artofproblemsolving.com/f/3/a/f3ac860e90d671f9fb592a6bd55c102a3000daec.png)
where 
is greatest integer not greater than 
of people, we define 
to be the least number of tables needed such that each person from the group sits in one of them with no two friends sitting on the same table, and 
to be the least number of tables needed such that each person from the group sits in one of them with no two enemies sitting on the same table.
such that for each group 
people we have![\[ f(n) \leqslant f(a(G)) + f(b(G)) \quad \text{and} \quad f(2) = 1.\]](http://latex.artofproblemsolving.com/0/3/d/03dfc700f9284152b39e32031c92c33283d65da6.png)
Find all possible values of 
.
is the set of all positive integers.
such that,
for all non-negative integers 
and 
.
is the set of all non-negative integers and 
is the set of all real numbers.
cells. It is said to be primitive it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.
cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.
such that 
and 
are coprime if and only if 
?
are positive reals. Consider the following polynomial for each 
in 
:
where indices are taken modulo 
,
for any 
in 
.
are written on a board. In each move, two numbers are selected such that their difference is also present on the board. This difference is then erased from the board. (For example, if the numbers 
and 
are on the board, then 
can be erased as 
,
as 
,
as 
.
and 
such that 
and 
is a prime number
at point 
intersect 
at 
is tangent to 
,
.
be real numbers such that 
.
Y by NO_SQUARES, ehuseyinyigit
Let 
be a triangle with incircle 
, which touches sides 
and 
at points 
and 
, respectively. Choose points 
and 
on the line 
such that 
and 
. Let 
be the intersection of lines 
and 
. Define 
and 
as the intersections of 
and 
with lines 
and 
, respectively. Assume that 
is the intersection of 
and 
. Prove that 
.
be a triangle with incircle 
, which touches sides 
and 
at points 
and 
, respectively. Choose points 
and 
on the line 
such that 
and 
. Let 
be the intersection of lines 
and 
. Define 
and 
as the intersections of 
and 
with lines 
and 
, respectively. Assume that 
is the intersection of 
and 
. Prove that 
.Attachments:
at 
.
is the point such that 
ia a parallelogram. 
intersect 
at 
,
intersect 
at 
.
bisect 
.
bisects segment 
,
