[/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 
.
its 
-
denote the midpoint of arc 
.
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 
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
Point 
is the center of the inscribed circle of triangle 
. A circle with center at intersects side 
at points 
and 
, side 
at points 
and 
, side AB at points 
and 
. Received points are located on the circle in the order , , , , , . Points 
, 
, 
are the midpoints of arcs 
, 
, 
respectively. Lines 
and 
intersect at point 
, lines 
and 
intersect at point 
, and lines 
and 
intersect at the point 
. Prove that the segments 
, 
, 
intersect at one point.
(S. Berlov)
is the center of the inscribed circle of triangle 
. A circle with center at intersects side 
at points 
and 
, side 
at points 
and 
, side AB at points 
and 
. Received points are located on the circle in the order , , , , , . Points 
, 
, 
are the midpoints of arcs 
, 
, 
respectively. Lines 
and 
intersect at point 
, lines 
and 
intersect at point 
, and lines 
and 
intersect at the point 
. Prove that the segments 
, 
, 
intersect at one point.(S. Berlov)
This post has been edited 1 time. Last edited by parmenides51, Apr 30, 2022, 7:06 AM
