[/quote]
,
has an incenter 
Suppose 
.
be the midpoint of 
.
.
meets 
again at point 
.
and 
be such that 
and 
.
lies on the plane such that 
is a parallelogram, and suppose the angle bisectors of 
and 
concur on 
.
and 
meet 
at 
and 
.
.
be a triangle. Let 
and 
be the intersections of the 
and 
angle bisectors with the opposite sides. Let 
.
where 
is the major 
arc midpiont.
and 
again at 
or 
and 
concur on 
Choose a number of the form 
,
a non-negative integer, and place it in an empty square.
Choose two (not necessarily consecutive) squares containing the same number, say 
and erase the number in the other square.
,
and circumcentre 
.
and 
lie on 
and 
.
meet 
at 
meet 
and 
have an intersection lie on line 
.
and![\[ f(x+y)+f(x^2+f(y))=f(f(x))^2+f(x)+f(y)+y,\forall x,y\in\mathbb{R}\]](http://latex.artofproblemsolving.com/c/8/9/c895ae7fdf8d7f284ac9fc94cc077d6edad6cbf0.png)
be a convex hexagon with 
and 
.
and 
.
and 
are the centers of mass of 
and 
,
.
.
the reflections of 
and 
,
are collinear if and only if 
coincide, prove that 
given that there exist positive integers 
satisfying
and 
as follows:
where 
are all the positive divisors of 
.
Find all positive integers 
.
Find all positive integers 
.
be an odd prime number. How many 
of 
are there, the sum of whose elements is divisible by 
be real numbers satisfying
a variable point on segment 
.
is constructed such that 
and 
.
and 
.
board, two opposite sides have been joined, forming a cylinder. Determine whether it is possible to place 
.
.
Y by
Let 
be the final value obtained after repeatedly summing the digits of 
until a single-digit number is reached. (For example: 
, because the digit sum of 
is 
and the digit sum of is 
). Evaluate the sum:
Proposed by Lia Chitishvili, Georgia
be the final value obtained after repeatedly summing the digits of 
until a single-digit number is reached. (For example: 
, because the digit sum of 
is 
and the digit sum of is 
). Evaluate the sum:
Proposed by Lia Chitishvili, Georgia
,
,
,
,
,
is the remainder when 
and 
,
and multiply our result by 
.
.
