AoPS Academy
board black or white so that the cells of each color form a polygon. Then Bob dissects the board into dominoes (rectangles consisting of two cells). Alice wants to maximize the number of two-colored dominoes, and Bob wishes to minimize it. What maximal number of two-colored dominoes can be guaranteed by Alice regardless of Bob’s moves? (Recall that the boundary of a polygon is a closed broken line without self-intersections.)
. If![\[|a+b+c|=|ab+bc+ca|=|abc|=1,\]](http://latex.artofproblemsolving.com/b/6/7/b6762ca030e77f37d385db6ebf0de03cff3799f5.png)
show that 
.
, 
, 
of an acute-angled triangle 
intersect at point 
. The bisectors of angles 
and 
of triangle 
meet the segments 
and 
at points 
and 
respectively. Denote the value of the angle 
by 
. Define 
and 
similarly. Find the sum 
.
with incircle 
tangent to 
at points 
, respectively. Let 
be a point such that its isogonal conjugate lies on the line 
(where 
is the circumcenter and 
the incenter of ). The line 
intersects segments 
and 
at points 
and 
, respectively, such that the circle with diameter 
meets 
at points 
and 
.
is tangent to the incircle at some point .
corresponding to vertices 
. Prove that the lines 
are concurrent.
In the 
plane with the origin , the line 
intersects the lines 
, and 
at the points 
, respectively. Find the maximum value of the area of 
plane we have the points 
. The area of 
as a function of 
is represented by:
(i);
(ii) (since 
);
(a maximum).
.
