,
is a non-negative integer, 
is an odd number, and let 
.
and for any positive odd number 
,
is a multiple of 
.
,
such that the inequality :![\[f(x) \leq f(x)^3-2f(x)^2+2\]](http://latex.artofproblemsolving.com/1/2/c/12c4c4c850c1c2886f002dfcb0bead075e2ab679.png)
.
,
for all 
and 
Prove that 
be real numbers such that 
.
Prove that 
and ![$e\in [-2,2]$](http://latex.artofproblemsolving.com/b/b/7/bb78c9dedb18af5e6a904f5492917921e41a7ebc.png)
such that 
Find the value of 
:
be the altitude and 
be the intersection point of the altitudes of triangle 
.
is symmetric to 
.
intersects 
and 
.
.
If the value of 
can be written as 
where 
and 
where 
and 
where 
.
lie on the same circle. If 
and 
,
,
or 11. However, I didn't know how to find the radius. Can anyone give me a hint?
then find the value of 
be a sequence of reals such that 
and 
for all positive integers 
approaches the real number 
.
Y by Adventure10, Mango247, and 2 other users
Some time ago I have tried to prove Hall's matching/marriage theorem on my own. As a result, I obtained a rather tedious linear-algebraic proof which I don't deem particularly interesting on its own, but it contains an idea that I would love to see applied at some more serious questions.
The proof can be found in a note called "An algebraic approach to Hall's matching theorem" on my website. The note itself is a pain to read and is just there to make sure that the proof has no flaws. If you are interested in the proof, try out the shortened version of the note, which is still 7 pages long (the first 2 of them being prerequisites, however). Given the fact that the Hall theorem can be shown on half a page (though I don't find the proofs quite natural, honestly), this is not quite a sensation.
Anyway, as I said, what matters is the idea, which is basically to fill an adjacency matrix or another kind of characteristic matrix not with
's and 
's but with 's and free variables. In the case of the Hall theorem, we have a bipartite graph with 
blue vertices 
, 
, ..., 
and green vertices 
, 
, ..., 
. (Have you guessed that B and G stand for blue and green?) We take an arbitrary field 
, for instance 
, and denote by 
the field of all rational functions of 
indeterminates 
, 
, ..., 
(one indeterminate 
for each pair 
) over . Then, we consider the matrix 
which has, in its 
-th row and 
-th column, the entry 
. This matrix "represents" the bipartite graph and has the nice property that it is invertible if and only if the bipartite graph has a perfect matching. My proof of Hall's theorem exploits the properties of this matrix (alas, in a rather ugly way).
Note that
is not an adjacency matrix, but we could easily define a similar variation on the concept of adjacency matrices, or incidence matrices. It is less clear to me, however, under which circumstances such variations can be of use. Therefore two questions:
- Is the idea/trick of filling characteristic matrices with free variables new?
- Can anyone find a less simple application of the trick?
Darij
The proof can be found in a note called "An algebraic approach to Hall's matching theorem" on my website. The note itself is a pain to read and is just there to make sure that the proof has no flaws. If you are interested in the proof, try out the shortened version of the note, which is still 7 pages long (the first 2 of them being prerequisites, however). Given the fact that the Hall theorem can be shown on half a page (though I don't find the proofs quite natural, honestly), this is not quite a sensation.
Anyway, as I said, what matters is the idea, which is basically to fill an adjacency matrix or another kind of characteristic matrix not with
's and 
's but with 's and free variables. In the case of the Hall theorem, we have a bipartite graph with 
blue vertices 
, 
, ..., 
and green vertices 
, 
, ..., 
. (Have you guessed that B and G stand for blue and green?) We take an arbitrary field 
, for instance 
, and denote by 
the field of all rational functions of 
indeterminates 
, 
, ..., 
(one indeterminate 
for each pair 
) over . Then, we consider the matrix 
which has, in its 
-th row and 
-th column, the entry 
. This matrix "represents" the bipartite graph and has the nice property that it is invertible if and only if the bipartite graph has a perfect matching. My proof of Hall's theorem exploits the properties of this matrix (alas, in a rather ugly way).Note that
is not an adjacency matrix, but we could easily define a similar variation on the concept of adjacency matrices, or incidence matrices. It is less clear to me, however, under which circumstances such variations can be of use. Therefore two questions:- Is the idea/trick of filling characteristic matrices with free variables new?
- Can anyone find a less simple application of the trick?
Darij
This post has been edited 1 time. Last edited by darij grinberg, Nov 18, 2008, 12:56 PM
