,![\[
\sqrt{a + b} + \sqrt{b + c} + \sqrt{c + a} \geq \frac{4(ab + bc + ca)}{\sqrt{(a + b)(b + c)(c + a)}}
\]](http://latex.artofproblemsolving.com/e/7/2/e72fd0f134289c359bc8766440babc52ae131961.png)
be real numbers such that 
Prove that
Let 
Prove that
Y by Adventure10, Mango247
Let 
, 
, 
, and 
be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex , observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let 
be the probability that the bug is at vertex when it has crawled exactly 7 meters. Find the value of 
.
, 
, 
, and 
be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex , observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let 
be the probability that the bug is at vertex when it has crawled exactly 7 meters. Find the value of 
.
be the probability that the bug is on vertex 
after 
moves. Clearly, this means 
and 
We can also define the recursion 
, because on the move prior to move 
we can repeat this recursion up to 
doing so we get 
and finally 
So 
as follows:![\[ P=\left[
\begin{array}{cccc}
0 & 1/3 & 1/3 & 1/3 \\
1/3 & 0 & 1/3 & 1/3 \\
1/3 & 1/3 & 0 & 1/3 \\
1/3 & 1/3 & 1/3 & 0
\end{array}
\right],\]](http://latex.artofproblemsolving.com/a/4/d/a4dc2e629a002b55460d1216f0714c86f36d06f3.png)
![$ \mathbf{x}^{(0)}=\left[
\begin{array}{c}
1 \\
0 \\
0 \\
0
\end{array}
\right]$](http://latex.artofproblemsolving.com/4/6/2/4622cd66af1f90c051f6eeb8c2f4c7382179e85a.png)
.
.
.![$ \mathbf{x}^{(7)}=\left[
\begin{array}{c}
182/729 \\
547/2187 \\
547/2187
\end{array}
\right]$](http://latex.artofproblemsolving.com/9/5/1/9511c5936550fc6cf0453fe908eaf73e277fa4fc.png)
.
.
steps you return to the same state as the state in which you started is the sum of the 
powers of the eigenvalues of 
:
.
,
denote the number of walks of length 
to vertex 
and let ![$ A[ij]$](http://latex.artofproblemsolving.com/7/4/c/74c1ba39a082d9eb849c034fbe148f48e95fd84c.png)
denote the matrix obtained from 
column and 
row. Then![$ \sum_{n \ge 0} a_{ij}(n) \lambda^n = \frac {( - 1)^{i + j} \det(I - \lambda A[ij]) }{\det(I - \lambda A) }$](http://latex.artofproblemsolving.com/7/a/5/7a569c0f128eb0ac795f88deb81657db63388aa2.png)
.
chance of going to A.
integers 
are there such that 
for all 
and 
for all 
?
Let 
be a new sequence of integers such that 
for all 
for all 
and since 
must be a multiple of 
such that 
s
ways to arrange the 
valid ways (the proof of this combinatorial identity is left as an exercise to the reader) to arrange the 
s
s
ways to arrange the 
valid ways to arrange the 
ways to arrange the 
valid ways to arrange the 
valid sequences. Dividing by the total number of paths without restrictions, 
our desired probability is 
giving an answer of 
