[/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)
and orthocenter 
,
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 
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)
.
satisfy that: 
and 
.
and 
Prove that
Y by Adventure10, Mango247
Given 3 non-collinear points 
. For each point 
in the plane (
) let 
be the point symmetric to with respect to 
, 
be the point symmetric to with respect to 
and 
be the point symmetric to with respect to 
. Find all points such that 
obtains its minimum. Let this minimum value be 
. Prove that does not depend on the order of the axes of symmetry we chose (we have 3 available axes, that is , 
, . In the first part the order of axes we chose , , , and the second part of the problem states that the value doesn't depend on this order).
. For each point 
in the plane (
) let 
be the point symmetric to with respect to 
, 
be the point symmetric to with respect to 
and 
be the point symmetric to with respect to 
. Find all points such that 
obtains its minimum. Let this minimum value be 
. Prove that does not depend on the order of the axes of symmetry we chose (we have 3 available axes, that is , 
, . In the first part the order of axes we chose , , , and the second part of the problem states that the value doesn't depend on this order).
