be the median through 
be the incircle. Let 
at point 
and segment 
meet 
.
be the triangle formed by lines 
be real numbers such that 
Prove that
Where 
Where 
are are the lengths of the sides of a 
g
then![$$(n-2)\left[\sum_{i=1}^{n}{\frac{a_i^2}{(1-a_i)^2}}-\frac n{(n-1)^2}\right]\geq(2n-1)\left(\sum_{i=1}^{n}{\frac{a_i}{1-a_i}}-\frac n{n-1}\right)^2.$$](http://latex.artofproblemsolving.com/9/3/e/93e5529518807b92cd9e0382f5fb07a293128b04.png)
and 
be a foot of the altitudes from 
and 
and let 
.
cuts 
in half.
and 
.
be an irrational number. Show that for every integer 
,
such that 
.
be positive integers with 
of the set of 
”
”
Y by PikaPika999
In the plane of traingle 
we consider a variable point 
which moves on line 
towards 
. Halfway there, it stops and starts moving in a straight line line towards 
. Halfway there, it stops and starts moving in a straight line towards 
, and halfway there it stops and starts moving in a straight line towards , and so on. Show that will get as close as we want to the vertices of a fixed triangle with area 
.
we consider a variable point 
which moves on line 
towards 
. Halfway there, it stops and starts moving in a straight line line towards 
. Halfway there, it stops and starts moving in a straight line towards 
, and halfway there it stops and starts moving in a straight line towards , and so on. Show that will get as close as we want to the vertices of a fixed triangle with area 
.
