, 2011

Observe that
$$\begin{aligned} S &> \frac{a}{a+b+c+d} + \frac{b}{a+b+c+d} + \frac{c}{a+b+c+d} + \frac{d}{a+b+c+d} = 1, \\ S &< \frac{a}{a+b} + \frac{b}{a+b} + \frac{c}{c+d} + \frac{d}{c+d} = 2. \end{aligned}$$
The function changes smoothly as we vary $a$, $b$, $c$ and $d$. We will prove that it comes arbitrarily close to $1$ and $2$, and therefore it assumes every value in the interval $(1,2)$.

Taking $a = b$ and letting $c = d$ gives
$$S_1(a, c) = \frac{2a}{2a+c} + \frac{2c}{a+2c}, \quad \Rightarrow \quad \lim_{c \to 0} S_1(a, c) = 1.$$
Similarly, taking $a = c$ and $b = d$ gives
$$S_2(a, b) = \frac{2a}{a+2b} + \frac{2b}{2a+b} \quad \Rightarrow \quad \lim_{b \to 0} S_2(a, b) = 2.$$
So the expression takes on all values between $1$ and $2$.