![\[
f(x) = \sqrt{1 - x^a}, \quad a \geq 0
\]](http://latex.artofproblemsolving.com/e/f/1/ef14689f1894683256f9a0155037788366496929.png)
![\( x \in [0,1] \)](http://latex.artofproblemsolving.com/7/6/d/76d950c78d04a107e9fcb49b26ab353531c93b92.png)
.
,
is a power series for which each coefficient 
is 
or 
.
,
must be irrational.
.
s.t. 
.
,![$K=\mathbb{Q}[\cos(\dfrac{2}{3}\arctan(3\sqrt{3}))]$](http://latex.artofproblemsolving.com/a/2/e/a2efd64f27687f9ba5239cc4e0ba49d469d1c8c9.png)
.
.
and not its conjugate transpose.
.
.
;
, ∀x∈[0,1].
be a finite abelian group. There is a magic box 
.
contains only the group identity, 
.
![$x\in [0,4]$](http://latex.artofproblemsolving.com/a/8/9/a89e045d4c66cdb788e64e613bd6c3c9b5931cc2.png)
at which the function 
takes its maximum value
Y by
Let 
be a positive real number. Divide the positive real axis into intervals 
, 
, 
, 
, 
, and color them alternately black and white. Consider the function 
satisfying the following differential equations:
![\[
u''(x) + 9^2u(x) = 0, \quad \text{for } x \text{ in black intervals},
\]](//latex.artofproblemsolving.com/2/6/8/268b7bc6150b13eed666d566b540df7e643d8344.png)
![\[
u''(x) + 63^2u(x) = 0, \quad \text{for } x \text{ in white intervals},
\]](//latex.artofproblemsolving.com/b/4/6/b46d0d00b34cd3fb72ec310625384547b14871c5.png)
with the initial conditions:
![\[
u(0) = 1, \quad u'(0) = 1,
\]](//latex.artofproblemsolving.com/4/1/6/416754efbff23b7efbf02b32149b1e1fd5cb9905.png)
and the continuity conditions:
![\[
u(x) \text{ and } u'(x) \text{ are continuous functions}.
\]](//latex.artofproblemsolving.com/b/3/0/b309f7b8c70cb27683a96af8452ebf9e36f67a4c.png)
Show that
![\[
\lim_{\omega \to 0} u\left(\frac{\pi}{45}\right) = 0.
\]](//latex.artofproblemsolving.com/f/0/b/f0b9105523f0561e0a4d151a2695cbf6917e5cc7.png)
be a positive real number. Divide the positive real axis into intervals 
, 
, 
, 
, 
, and color them alternately black and white. Consider the function 
satisfying the following differential equations:![\[
u''(x) + 9^2u(x) = 0, \quad \text{for } x \text{ in black intervals},
\]](http://latex.artofproblemsolving.com/2/6/8/268b7bc6150b13eed666d566b540df7e643d8344.png)
![\[
u''(x) + 63^2u(x) = 0, \quad \text{for } x \text{ in white intervals},
\]](http://latex.artofproblemsolving.com/b/4/6/b46d0d00b34cd3fb72ec310625384547b14871c5.png)
with the initial conditions:![\[
u(0) = 1, \quad u'(0) = 1,
\]](http://latex.artofproblemsolving.com/4/1/6/416754efbff23b7efbf02b32149b1e1fd5cb9905.png)
and the continuity conditions:![\[
u(x) \text{ and } u'(x) \text{ are continuous functions}.
\]](http://latex.artofproblemsolving.com/b/3/0/b309f7b8c70cb27683a96af8452ebf9e36f67a4c.png)
Show that![\[
\lim_{\omega \to 0} u\left(\frac{\pi}{45}\right) = 0.
\]](http://latex.artofproblemsolving.com/f/0/b/f0b9105523f0561e0a4d151a2695cbf6917e5cc7.png)
