[/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)
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 
.
and 
be the points on the circumcircle of triangle 
is a diameter, 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 
.
.
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, Mango247
Let 
be an acute triangle, which is not equilateral. Denote by 
and 
its circumcenter and orthocenter, respectively. The circle 
passes through 
and touches the line 
at 
. The circle 
with center on the ray 
touhes the line 
at . The circles and meet in 
(
). Show that 
.
Proposed by Josef Tkadlec
be an acute triangle, which is not equilateral. Denote by 
and 
its circumcenter and orthocenter, respectively. The circle 
passes through 
and touches the line 
at 
. The circle 
with center on the ray 
touhes the line 
at . The circles and meet in 
(
). Show that 
.Proposed by Josef Tkadlec
and 
,
,
.
=
.
,
.
be the reflection of 
.
and 
are similar, hence 
.
by 
criterion. This implies that 
and 
are similar too, getting the desired result.
be the centers of 
(mod 
)
(1)
(mod 
(3)
(same direction)
