[/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 
-
denote the midpoint of arc 
.
meets 
.
at 
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?
,
and that line 
is not parallel to line 
.
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 
.
.
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
On the sides 
and 
of a triangle 
points 
and 
, respectively, are taken. The circumcircle of the triangle meets the line through 
parallel to 
in a point 
(other than ). The circumcircle of the triangle 
meets the line through parallel to 
in a point 
(other than ). Prove that the lines 
, and 
are concurrent.
and 
of a triangle 
points 
and 
, respectively, are taken. The circumcircle of the triangle meets the line through 
parallel to 
in a point 
(other than ). The circumcircle of the triangle 
meets the line through parallel to 
in a point 
(other than ). Prove that the lines 
, and 
are concurrent.This post has been edited 2 times. Last edited by parmenides51, Mar 16, 2021, 7:23 PM
Reason: title typo
Reason: title typo
.
.
.
are collinear. Let 
.
is cyclic. Finally we have 
are collinear. We are done.
