[/quote]
,
be a positive integer. Alex plays on a row of 9 squares as follows. Initially, all squares are empty. In each turn, Alex must perform exactly one of the following moves:
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 
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 
.
which is equidistant from sides 
and 
.
and 
are the centers of mass of 
and 
,
.
.
the reflections of 
and 
,
are collinear if and only if 
coincide, prove that 
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
be a fixed square and 
a variable point on segment 
.
is constructed such that 
and 
be the intersection point of lines 
and 
.
board, two opposite sides have been joined, forming a cylinder. Determine whether it is possible to place 
.
.
denote the number of positive integers less than 
has at least 
solutions in 
that can be written in the form
where 
are pairwise coprime positive integers such that 
,
,
.
Y by
In triangle 
, let 
and 
be the circumcenter and orthocenter, respectively. Let 
and 
be the midpoints of 
and 
, respectively, and let 
and 
be the feet of the perpendiculars from 
and 
to the opposite sides, respectively. Show that if 
is the intersection of 
and 
, then 
is perpendicular to 
.
, let 
and 
be the circumcenter and orthocenter, respectively. Let 
and 
be the midpoints of 
and 
, respectively, and let 
and 
be the feet of the perpendiculars from 
and 
to the opposite sides, respectively. Show that if 
is the intersection of 
and 
, then 
is perpendicular to 
.
