AoPS Academy
and 
. Define 
for 
as 
, 
. Prove that 
exists and find it's value .
be a function that satisfies the conditions:
has Darboux’s property
at 
exist.
![\[ \int_{1}^{2}e^{x( x+\sqrt{x^2-1} )}dx \]](http://latex.artofproblemsolving.com/b/1/d/b1d1f5525bb9cb5fe1f1a7491ff5209cad5438d3.png)
such that
. Show that 
is an increasing function on 
, and 
.
, and 
for 
using the same 
and 
axes.
.
.
; if the answer is yes, then give a method to calculate at least one of its roots.
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 
?
be a prime number. A flea is staying at point 
of the real line. At each minute,
to the left or to the right.
minutes, it wants to be at again. Denote by 
the number of its strategies to do this
: it may either stay at for the entire time, or go to the left and then to the modulo .
be a continuous function such that for all 
and for all 
, 
Show that is a constant function.
here is the top 5 scores 39-39-40-41-41 so I mean it just matters if you practice enough.
