AoPS Academy
+1 w
unit can be expressed in the form 
units
, where 
and 
are integers. What is the value of 
?
and 
for all 
in the domain of 
, what is the value of 
?
.
s.t. 
.
, where ![$K=\mathbb{Q}[\sin^2(\dfrac{2}{3}\arctan(3\sqrt{3}))]$](http://latex.artofproblemsolving.com/5/b/8/5b87dd74c6659350c000071d5a9228f2a80b1466.png)
.
.
and not its conjugate transpose.
be a finite abelian group. There is a magic box 
. At any point, an element of may be added to the box and all elements belonging to the subgroup (of ) generated by the elements currently inside are moved from outside to inside (unless they are already inside). Initially 
contains only the group identity, 
. Alice and Bob take turns moving an element from outside to inside it. Alice moves first. Whoever cannot make a move loses. Find all for which Bob has a winning strategy.
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)
![$x\in [0,4]$](http://latex.artofproblemsolving.com/a/8/9/a89e045d4c66cdb788e64e613bd6c3c9b5931cc2.png)
at which the function 
takes its maximum value
be an algebraic extension of 
, and let 
be an algebraic closure of containing . Prove that if 
is an embedding such that 
for all 
, then 
extends to an automorphism of . is an algebraic extension of the field , is an algebraically closed field, and 
is an embedding. Then, there exists an embedding 
that extends . Moreover, if is an algebraic closure of and is an algebraic extension of 
, then 
is an isomorphism.
and is an algebraic extension of , it follows that is an algebraic extension of . Assume that there exists an embedding such that for all . By Theorem (*), there exists an embedding 
that extends . Since is algebraically closed, 
is also an algebraically closed field. for all and is an extension of , we have
is an algebraic extension of . We conclude that 
, meaning that is an automorphism. (Finished!!)
be the field of algebraic numbers, 
. Consider the embedding 
defined by
Then, according to the exercise above, extends to an isomorphism
How should we interpret 
?
, let 
. Find all convex sets 
such that![\[\mathcal{E}(B \cup C) = B \cup C.\]](http://latex.artofproblemsolving.com/e/c/3/ec34115c9e6e169c19c739220065f647934a1a5f.png)
