[/quote]
,
.
.
is the intersection of the parallel line from 
with the perpendicular line to 
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?
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)
,
be an arbitrary point on the circumcircle of triangle 
and that line 
is not parallel to line 
.
be the point on the circumcircle of triangle 
is perpendicular to 
be the point on line 
.
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 
.
.
be positive real numbers such that 
.![\[ \frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_{1})(1 - a_{2}) \cdots (1 - a_{n})} \leq \frac{1}{ n^{n+1}}. \]](http://latex.artofproblemsolving.com/6/5/5/65558feba82e32266d3d3cbdfea85e079483403f.png)
with integer coefficients such that for all positive number 
and prime 
satisfying 
,
.
be reals such that 
P
,
,
is __
satisfying 
Y by Adventure10, Mango247
Let 
be an interior point of the triangle 
such that 
, 
, and 
. The circumcenters of the triangles 
, 
, and 
are 
, 
, and 
respectively. Prove that the area of 
is greater than or equal to the area of 
.
be an interior point of the triangle 
such that 
, 
, and 
. The circumcenters of the triangles 
, 
, and 
are 
, 
, and 
respectively. Prove that the area of 
is greater than or equal to the area of 
.This post has been edited 1 time. Last edited by Karth, Apr 23, 2006, 5:53 PM
.
,![$[ABD] + [CBE] \ge 2 [ABC]$](http://latex.artofproblemsolving.com/b/0/2/b02527f145614bc87c9a9a13b52673f3c1cc3e79.png)
.
EDITED.
?
