[/quote]
,
board, each square is colored either white or black, with all the squares on the border of the board being black. Additionally, no 
square within the board has all four squares of the same color. Prove that the board contains a 
,
,
is __
prove that![$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$](http://latex.artofproblemsolving.com/6/a/6/6a657cb647e279b0745e3189c3f9d54aa25bf976.png)
satisfying 
.
denote its incenter and 
its 
.
meets 
.
at 
and 
at 
respectively. The circumcircle of 
meets 
a second time at 
meets 
a second time at 
passes through the antipodal to 
.
.
is the intersection of the parallel line from 
.
be the point where the external bisector of 
intersects with 
be the projection of 
.
have a common point.
dwarves, whose weigths are 
,
each time without sinking. For each journey of the boat a non-empty subset of the dwarves to be taken to the other side is chosen and one of these dwarves is chosen as the 
for that journey. Since return journeys are counter-current, no dwarf is able to do the rower for more than one return journey. Is it possible for all the dwarves to reach the other side of the river?
,
and that line 
is not parallel to line 
is perpendicular to 
.
be the points on the circumcircle of triangle 
is a diameter, and 
and 
are parallel. Let 
.
and 
are perpendicular.
and 
are perpendicular.
with 
,
intersects the extension of 
at point 
.
is the midpoint of 
,
intersects the circumcircle of 
.
.
Y by Adventure10
Given an acute triangle 
with 
. The angle bisector of 
intersects 
and 
at 
and 
respectively. 
is on the angle bisector of such that is between 
and . 
and 
are on such that 
. If 
and 
intersects the second time at point 
.
Prove that
are collinear if and only if is the incenter of 
.
with 
. The angle bisector of 
intersects 
and 
at 
and 
respectively. 
is on the angle bisector of such that is between 
and . 
and 
are on such that 
. If 
and 
intersects the second time at point 
.Prove that
are collinear if and only if is the incenter of 
.
mixtilinear circle with 
.
