[/quote]
,
be an 
real matrix with minimal polynomial 
.
is zero.
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
.![$\sqrt[3]{ab}+ \sqrt[3]{cd} < \sqrt[3]{(a+b+c)(b+c+d)}$](http://latex.artofproblemsolving.com/0/5/0/0502ba65f0545b3af84c41b2549d6256550b473b.png)
.
be triangle with circumcenter 
and orthocenter 
.
intersect 
respectively. Lines through 
intersects 
at 
respectively. Point 
such that 
is a parallelogram. Prove that lines 
and 
are concurrent at a point on 
.
satisfying
Prove
and 
be positive integers. Suppose that the sum of integers between 
.
,
![$C[a,b]$](http://latex.artofproblemsolving.com/e/e/6/ee63b22d81304485423eec80b34bbf21b655d75d.png)
is the set of all continuous functions on the closed interval ![$[a,b]$](http://latex.artofproblemsolving.com/8/e/c/8ecbd1ba3da8f2adef66a63f2ab32c47e63fa734.png)
.![$$d(f, g) = \sup_{x \in [a, b]} |f(x) - g(x)|.$$](http://latex.artofproblemsolving.com/a/4/a/a4a62b37713788c94992853b0eaadcbabbca86d6.png)
(b) Consider the closedness and compactness of the following set in the space :![$$B = \{ f \in C[a, b] : |f(x)| \leq 1, \text{ for all } x \in [a, b] \}.$$](http://latex.artofproblemsolving.com/2/e/d/2ed13593e5bad1171aad457042dcdc9149d9d931.png)
.
be the circumcircle of 
be the midpoint of the arc 
of 
meets 
at 
.
at 
.
by 
.
.
meets line 
.
and 
be two lines through the points 
and 
,
.
,
intersect 
and let 
intersect 
.
,
and 
are circumcenters of the triangles 
and 
,
,
.
is 
moves on sides 
,
.
is a definite value.
be the open unit disk in the complex plane and let 
be a holomorphic function such that 
.
be 
.
.
.
Where 
.
Y by
Q) let 
and 
then is this uniformly convergence on 
comment on uniformly convergence on ![$[0,1]$](//latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
where in general it is should be uniformly convergence.
My I am trying with some contradicton method like chose
and trying to solve
Next take a in (0,1) and chose a= 2^1/N but not solution
How to solve like this way help.
and 
then is this uniformly convergence on 
comment on uniformly convergence on ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
where in general it is should be uniformly convergence.My I am trying with some contradicton method like chose
and trying to solve
Next take a in (0,1) and chose a= 2^1/N but not solution
How to solve like this way help.
.![$[0,a]$](http://latex.artofproblemsolving.com/6/8/b/68bacdc4b96052586744d16169ccd8c3233ed09f.png)
for any 
because 
.
