and 
on 
and 
are on the same side of 
be midpoint of segment 
be midpoint of segment 
be intersection of 
with 
and 
be two positive integers, such that 
.
can take, where ![\[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]](http://latex.artofproblemsolving.com/c/b/2/cb27ccbcde6d90f98e10f8f6876c36ae8972af7f.png)
be positive real integer such that 
Prove that
such that 
and 
real number on the board such that each number is more than 
.
on the board and write 
.
and 
.
?
of positive integers satisfying the equation 
be a natural number . Define 
be cut number at 
if
Prove that if 
be cut number at 
prove that 
.
be 
altitude of 
.
and 
such that 
and 
. Point 
and 
. Point 
and 
intersect circumcircle of 
.
.
bisect 
Define a cell with 
property if after remove that cell from the board The board can be tile with 
tetromino.
Note: 
Prove that
Y by
Let 
be the feet of the altitudes of the triangle 
from 
, respectively. 
is the point on the circumcircle of the triangle , 
is the orthocenter of the triangle , and the incircle of triangle 
has radius 
. Let 
be the point such that and are on the opposite side of 
, line 
is perpendicular to the line , and the distance from to line is . Define 
and 
similarly. Prove that 
are collinear.
be the feet of the altitudes of the triangle 
from 
, respectively. 
is the point on the circumcircle of the triangle , 
is the orthocenter of the triangle , and the incircle of triangle 
has radius 
. Let 
be the point such that and are on the opposite side of 
, line 
is perpendicular to the line , and the distance from to line is . Define 
and 
similarly. Prove that 
are collinear.
midpoint of 
.
are on nine-point circle of 
.
be midpoint of 
. so it sufficient to prove that 
be point that incircle of 
respectively.
be projection 
respectively.
