[/quote]
,
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 
.
meets 
.
at 
and 
at 
and 
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?
,
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 
.
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
Y by Adventure10, Mango247
Let 
, 
are excircles of a triangle 
. Given a circle 
passes through 
and externally tangents to the circles and such that it intersects with 
at points 
, 
.
Prove that
.
A. Smirnov
, 
are excircles of a triangle 
. Given a circle 
passes through 
and externally tangents to the circles and such that it intersects with 
at points 
, 
.Prove that
.A. Smirnov
This post has been edited 1 time. Last edited by mathuz, Apr 20, 2014, 2:00 PM
is also center of negative inversion that transforms them into each other. By conformity, 
is transformed into the common external tangent 
of 
different from 
The center of 
but 
it is on the A-circumdiameter of 
of 
Hence, if 
cut 
we have 
are equal 
