[/quote]
,
are non-negative real numbers and 
![\[\frac{a(a^2-bc)}{\sqrt{ka+b+c}} + \frac{b(b^2-ca)}{\sqrt{kb+c+a}} + \frac{c(c^2-ab)}{\sqrt{kc+a+b}} \geqslant 0.\]](http://latex.artofproblemsolving.com/5/f/9/5f9deb36a5aee41d3fa177b53391b7149fb76a6e.png)
(ii) Prove that![\[(a^2-bc)\sqrt{ka+b+c}+(b^2-ca)\sqrt{kb+c+a}+(c^2-ab)\sqrt{kc+a+b} \geqslant 0.\]](http://latex.artofproblemsolving.com/2/c/4/2c4e7a84750ebc1521a0190076b116ada67e3937.png)
be a point in the plane, and 
a line not passing through 
satisfying
Prove
.![$\sqrt[3]{ab}+ \sqrt[3]{cd} < \sqrt[3]{(a+b+c)(b+c+d)}$](http://latex.artofproblemsolving.com/0/5/0/0502ba65f0545b3af84c41b2549d6256550b473b.png)
.
.
such 
converges then there is some 
such that
and 
converges.
and 
points be given. Show that the vertices can be enumerated as 
such that 
matrix such that 
,
is a permutation of 
,
.
is a permutation of 
.
be an acute, non-isosceles triangle, 
be its heights and 
its circumcircle. 
cuts the circumcircle at points 
,
being between points 
.
be the midpoints of the major and the minor arc 
,
cuts 
.
and 
cut again 
at points 
and 
,
is the line passing through the circumcenters of triangles 
and 
the parallel lines through 
to 
meets 
at 
and 
meets 
at 
,
are concyclic.
such that: 
.![$$ \sqrt[3]{4} \geq a^2-ab+b^2 \geq 1$$](http://latex.artofproblemsolving.com/e/a/2/ea2874bcb691804ea1cc026a5cba87f04d17f64e.png)
derivable on 
with 
be two 
complex matrices of the same rank, and let 
be a positive integer. Prove that 
if and only if 
.
Y by
Given a n*n matrix A, prove that there exists a matrix B such that ABA = A
Solution: I have submitted the attachment
The answer is too symbol dense for me to understand the answer.
What I have undertood:
There is use of direct product in the orthogonal decomposition. The decomposition is made with kernel and some T (which the author didn't mention) but as per orthogonal decomposition it must be its orthogonal complement.
Can anyone explain the answer in much much more detail with less use of symbols ( you can also use symbols but clearly define it).
Also what is phi | T ?
Solution: I have submitted the attachment
The answer is too symbol dense for me to understand the answer.
What I have undertood:
There is use of direct product in the orthogonal decomposition. The decomposition is made with kernel and some T (which the author didn't mention) but as per orthogonal decomposition it must be its orthogonal complement.
Can anyone explain the answer in much much more detail with less use of symbols ( you can also use symbols but clearly define it).
Also what is phi | T ?
