[/quote]
,
are complex numbers with absolute value 
.
define 
.
converges. Show that,
.
![$g_i \in C([0,1],\mathbb R_+)$](http://latex.artofproblemsolving.com/3/7/9/379b9e3f56dd862fa6b403bf9b3f799ffc109b2d.png)
convex.![$a \in [0;1]$](http://latex.artofproblemsolving.com/a/f/b/afbcf487b151bb65405d3d385abf89b9393ce9af.png)
with 
?
be prime, let 
be a positive integer, show that ![\[ \gcd\left({p - 1 \choose n - 1}, {p + 1 \choose n}, {p \choose n + 1}\right) = \gcd\left({p \choose n - 1}, {p - 1 \choose n}, {p + 1 \choose n + 1}\right). \]](http://latex.artofproblemsolving.com/9/d/4/9d409ce186320241eb358b5fbbb8a5a66120899c.png)
be a right triangle with a right angle in 
and a circumcenter 
.
and 
,
and 
lie in such a way that 
.
and 
be the projection of 
,
is half as long as 
be an isosceles triangle with 
and circumcircle 
.
perpendicular to 
.
be any point on 
with center 
intersects 
and the circumcircle 
.
and 
,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
be the incentre of a non-equilateral triangle 
be the 
-
be the reflection of 
be the reflection of line 
in 
.
,
and line 
analogously. Let 
where 
is the circumcentre of triangle 
and 
.
.
such that 
is a perfect square and 
is a prime number.
,
,
,![\[ \frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} \geq a+b+c \]](http://latex.artofproblemsolving.com/9/f/a/9fa6cca20b3620fbb32ee782196e96f6fffa0063.png)
such that for any real value of 
we have:
,
,![\[ f(x)f(yf(x))=f(x+y). \]](http://latex.artofproblemsolving.com/5/6/0/560049dd70944cafa0423e5f1229d215f78215a0.png)
........
?
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 ?
