[/quote]
,
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)
with circumcenter 
and that line 
be the point on the circumcircle of triangle 
is perpendicular to 
be the point on line 
such that 
.
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 
.
.
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 
denote its inradius and ex-radii opposite to the vertices 
respectively. If 
and 
,
is obtuse
is acute
,
is semi perimeter
,![$ a,b\in[\frac{1}{2},1] $](http://latex.artofproblemsolving.com/0/d/2/0d24f742e1f7db553b3a8795c061c796f07a897f.png)
.
Let ![$ a,b\in[\frac{1}{2},2] $](http://latex.artofproblemsolving.com/a/8/d/a8d4c8205bac0e34129f582a8f255355eda12e0b.png)
.
Y by Adventure10
Let 
and 
be two fixed points on a circle centered at 
that are not diametrically opposed. Let 
be a variable point on the circle distinct from and and not belonging to the perpendicular bisector of 
. Let 
be the orthocenter of 
, and 
and 
be the midpoints of the segments and 
, respectively. The line 
intersects the circle again at 
, and finally, 
and 
intersect at 
. Determine the locus of points as moves around the circle.
and 
be two fixed points on a circle centered at 
that are not diametrically opposed. Let 
be a variable point on the circle distinct from and and not belonging to the perpendicular bisector of 
. Let 
be the orthocenter of 
, and 
and 
be the midpoints of the segments and 
, respectively. The line 
intersects the circle again at 
, and finally, 
and 
intersect at 
. Determine the locus of points as moves around the circle.
![$[i]r[/i]$](http://latex.artofproblemsolving.com/a/7/e/a7e18c283a412e3621c0c615a6f2948c17258c7c.png)
be the radius of the circle, 
and 
is a parallelogram and we also have 
is an isosceles triangle with ![$OA = OD = [i]r[/i]$](http://latex.artofproblemsolving.com/6/3/b/63bd583463a2a7073ae46dcb8081486d0ea7a94c.png)
)
is also an isosceles triangle and 
,
.
)
