[/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 
is inscribed in circle 
.
denote its incenter and 
its 
-
denote the midpoint of arc 
.
meets 
.
at 
and 
at 
and 
respectively. The circumcircle of 
meets 
a second time at 
,
meets 
a second time at 
.
passes through the antipodal to 
.
be a triangle with circumcenter 
.
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?
,
be an arbitrary point on the circumcircle of triangle 
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 
be the midpoint of 
.
and 
are perpendicular.
and 
are perpendicular.
with 
,
intersects the extension of 
at point 
.
is the midpoint of 
,
intersects the circumcircle of 
.
.
Y by godofgeometry, ATimo, Adventure10, Mango247
Let M be arbitrary point inside triangle ABC and N be arbitrary point of the segment AM.Straight lines AB and AC intersect the circumcircle of triangle BMC for the second time at points E and F respectively. Straight line EM intersects the circumcircle of triangle NMC for the second time at point P , while straight line FM intersects the circumcircle of triangle NMB for the second time at point Q. Prove that the circumcircles of triangles EMF and PMQ touch each other.
