AoPS Academy
has no solutions in integers 
, 
, and 
.
be a 
board. Some of its cells are colored black in such a way that every 
board of has at most 
black cells. Find the maximum amount of black cells that the board may have.
intersect at 
, and there exist a circle 
tangent to the extensions of 
at 
respectively. Circle 
passes through points 
, and is externally tangent to circle at 
. Prove that 
.
let 
be the foot of the altitude from 
to the side 
and 
, 
, 
be the incenter, -excenter, and 
-excenter, respectively. Denote by 
and 
the other intersection points of the circle 
with the lines 
and 
, respectively. Prove that 
.
,
and 
,
a right-angled triangle can be formed?
.
is the greatest side. Then, checking the right angle condition, 
.
and 
,
than the LHS, so the equality cannot hold.