be a triangle. The incircle of 
and 
at the points 
and 
be the point where the lines 
and 
meet, and let 
and 
and 
are parallelogram.
.
,
and 
be pairwise distinct nonnegative real numbers. Prove that![\[
(a + b + c) \left( \frac{a}{(b - c)^2} + \frac{b}{(c - a)^2} + \frac{c}{(a - b)^2} \right) > 4.
\]](http://latex.artofproblemsolving.com/2/0/9/209f9807481e7a1e97f023bf65a603cb462c62cb.png)
(Karl Czakler)
and 
of circle 
meet at point 
.
parallel to 
,
meets 
.
and let 
be the reflection of 
.
passes through the midpoint of 
.
of 
and the length of 
is 
intersects 
.
.
Y by Adventure10
Let 
be a square with sides length 
. Let 
be a path within which does not meet itself and which is composed of line segments 
with 
. Suppose that for every point 
on the boundary of there is a point of at a distance from no greater than 
. Prove that there are two points 
and 
of such that the distance between and is not greater than 
and the length of the part of which lies between and is not smaller than 
.
be a square with sides length 
. Let 
be a path within which does not meet itself and which is composed of line segments 
with 
. Suppose that for every point 
on the boundary of there is a point of at a distance from no greater than 
. Prove that there are two points 
and 
of such that the distance between and is not greater than 
and the length of the part of which lies between and is not smaller than 
.
is quite trivial in one direction, but if they ean the shortest distance, we just take points very close to the boundary and with angles close to 
degrees except in the corners, what is the mistake in those ideas?
.
to 
along 
from the current point. Obviously at any point the red points are a finite union of red segments (with possibly length zero). Suppose such points 
are opposite sides of squares, then we must completely paint 
.
,
and 
painted before 
was painted before 
