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
.
is an increasing function on 
,
.
,
for 
using the same 
and 
axes.
.
.
;
be an algebraic extension of 
,
be an algebraic closure of 
is an embedding such that 
for all 
,
extends to an automorphism of 
is an embedding. Then, there exists an embedding 
that extends 
,
is an isomorphism.
and 
that extends 
is also an algebraically closed field.
,
be the field of algebraic numbers, 
.
defined by
Then, according to the exercise above, 
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 
the number of its strategies to do this
:
be a continuous function such that for all 
and for all 
,
Show that 
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Hello guys,
I think we should start a factoring marathon. Post your solutions like this , and your problems like this . Please make your own problems, and I'll start off simple:
I think we should start a factoring marathon. Post your solutions like this , and your problems like this . Please make your own problems, and I'll start off simple:
This post has been edited 1 time. Last edited by pican, Nov 15, 2015, 12:40 AM
.
or 
.
.
.![$\dfrac{1}{2}\left[((a-b)(a+b))^2+((b-c)(b+c))^2+((c-a)(c+a))^2\right]$](http://latex.artofproblemsolving.com/1/3/4/13492404f04cf780c7485c2ccfd4ce209a93a14e.png)
?
?
or 
.
?
