Maximum value of expression: $\frac{ab+bc+cd}{a^2+b^2+c^2+d^2}$ What is the maximum value of 
$$\frac{ab+bc+cd}{a^2+b^2+c^2+d^2}$$ 
where $a,b,c$, and $d$ are real numbers?

Have a look at the Rayleigh quotient of two quadratic forms. This expression is homogeneous of degree zero in vector $(a,b,c,d)$ so you can look for its maximum on unit sphere.
Then if $X=(a,b,c,d)$ is on this sphere and 
$$Q = \frac{1}{2}\begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0\end{pmatrix}$$
You have
$$\frac{ab+bc+cd}{a^2+b^2+c^2+d^2} = \frac{X^T Q X}{X^TX}=X^T Q X$$
This is maximum for eigen-vector of $Q$ associated to its maximum eigen-value (which seems to be half of golden ratio $\frac{(1+\sqrt{5})}{4}$)
