[/quote]
,
is a complex matrix with 
prove that 
real matrix 
,
its counter-clockwise 
rotation.![$$
\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right]^R=\left[\begin{array}{lll}
3 & 6 & 9 \\
2 & 5 & 8 \\
1 & 4 & 7
\end{array}\right]
$$](http://latex.artofproblemsolving.com/4/b/8/4b8267cc7835e97d7f8a0b03e9e487f14522c1c0.png)
Prove that if 
then for any eigenvalue 
of 
or 
.
and circumscribed about circle 
.
respectively; line 
intersects 
at 
,
at 
.
be the tangency point of circle 
intersects line 
.
bisects 
.
for given positive reals 
.
for every 
.
is written on the board. Artur is the first to make a move. In each move, the player replaces the polynomial 
on the board with one of the following polynomials of their choice:
where 
is an arbitrary integer.
be the polynomial on the board after the game ends. Maryam wins if the equation 
has a finite and odd number of positive integer solutions 
Show that Maryam can always win the game, no matter how Artur plays.
such that
for all 
.
is inscribed in circle 
R
and 
intersect at 
is chosen on the diagonal 
so that 
is chosen on the segment 
so that 
Prove that line 
touches 
.
,
divide the circumcircle 
of the triangle 
,
be a variable point on the arc 
and 
be the incenters of the triangles 
and 
.
intersects the circle 
be a positive integer. Find the maximum number of diagonals in a regular 
the inequality holds: 
.
and let 
be the set 
.
divisible by 
such that if we define multi-set, 
then 
be such points on sides 
of a triangle 
is cyclic. 
,
is the midpoint of 
.
and 
let 
,
be the 
be the 
and 
.
be the number of pairs of integers 
such that the number 
is a perfect square. Prove that the limit
exists and find its value.
![\[
\lim_{x \to 0^+} \left\{ \lim_{n \to \infty} \left( \frac{\left\lfloor 1^2 (\sin x)^x \right\rfloor + \left\lfloor 2^2 (\sin x)^x \right\rfloor + \cdots + \left\lfloor n^2 (\sin x)^x \right\rfloor}{n^3} \right) \right\},
\]](http://latex.artofproblemsolving.com/5/b/f/5bf499eb70b5978e4b0e033e0706624ad341d6cc.png)
where 
denotes the floor function
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.
Is this is a good approach or any simple way please prefer.
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.
Is this is a good approach or any simple way please prefer.
