AoPS Academy
.
.
; if the answer is yes, then give a method to calculate at least one of its roots.
, ∀x∈[0,1].
.
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)
against time. We have 
. The area under the curve between 
and 
is the distance 
. But since the acceleration is monotonically decreasing, the curve is concave and hence the area under it is at least the area of the triangle formed by joining the origin to the point 
(other vertices 
and 
). Hence 
, so 
. This is achieved by a particle moving with constant acceleration.
. Displacement 
. By the condition, we are given that 
, 
and acceleration 
monotonically decreases. Then we obtain the integral equations
Let us use Chebyshev's inequality for monotone functions
a distance 