$ A$ takes $ m$ times as long to do a piece of work as $ B$ and $ C$ together; $ B$ takes $ n$ times as long as $ C$ and $ A$ together; and $ C$ takes $ x$ times as long as $ A$ and $ B$ together.  Then $ x$, in terms of $ m$ and $ n$, is:

$ \textbf{(A)}\ \frac {2mn}{m \plus{} n} \qquad \textbf{(B)}\ \frac {1}{2(m \plus{} n)} \qquad \textbf{(C)}\ \frac {1}{m \plus{} n \minus{} mn} \qquad \textbf{(D)}\ \frac {1 \minus{} mn}{m \plus{} n \plus{} 2mn} \qquad \textbf{(E)}\ \frac {m \plus{} n \plus{} 2}{mn \minus{} 1}$
1. Let \( A, B, \) and \( C \) respectively work at rates of \( a, b, \) and \( c \). Furthermore, let the total amount of work be \( 1 \). Then, we have the following relationships:
   \[
   \frac{1}{a} = m \cdot \frac{1}{b+c}
   \]
   \[
   \frac{1}{b} = n \cdot \frac{1}{a+c}
   \]
   \[
   \frac{1}{c} = x \cdot \frac{1}{a+b}
   \]

2. Take the reciprocals of each equation:
   \[
   a = \frac{b+c}{m}
   \]
   \[
   b = \frac{a+c}{n}
   \]
   \[
   c = \frac{a+b}{x}
   \]

3. Multiply all three equations:
   \[
   a \cdot b \cdot c = \left(\frac{b+c}{m}\right) \cdot \left(\frac{a+c}{n}\right) \cdot \left(\frac{a+b}{x}\right)
   \]

4. Simplify the right-hand side:
   \[
   abc = \frac{(b+c)(a+c)(a+b)}{mnx}
   \]

5. Expand the numerator:
   \[
   (b+c)(a+c)(a+b) = (b+c)(a^2 + ab + ac + bc)
   \]
   \[
   = a^2b + a^2c + ab^2 + abc + abc + b^2c + ac^2 + bc^2
   \]
   \[
   = a^2b + a^2c + ab^2 + 2abc + b^2c + ac^2 + bc^2
   \]

6. Substitute back into the equation:
   \[
   abc = \frac{a^2b + a^2c + ab^2 + 2abc + b^2c + ac^2 + bc^2}{mnx}
   \]

7. Divide both sides by \( abc \):
   \[
   1 = \frac{a^2b + a^2c + ab^2 + 2abc + b^2c + ac^2 + bc^2}{abc \cdot mnx}
   \]

8. Group the terms in the numerator:
   \[
   1 = \frac{2abc + a^2b + ab^2 + a^2c + ac^2 + b^2c + bc^2}{abc \cdot mnx}
   \]

9. Simplify the numerator:
   \[
   1 = \frac{2abc + abc \left(\frac{a}{b} + \frac{b}{a} + \frac{a}{c} + \frac{c}{a} + \frac{b}{c} + \frac{c}{b}\right)}{abc \cdot mnx}
   \]

10. Divide both sides by \( abc \):
   \[
   1 = \frac{2 + \frac{a}{b} + \frac{b}{a} + \frac{a}{c} + \frac{c}{a} + \frac{b}{c} + \frac{c}{b}}{mnx}
   \]

11. Group the RHS numerator by similar denominators:
   \[
   1 = \frac{2 + \frac{a+c}{b} + \frac{b+c}{a} + \frac{a+b}{c}}{mnx}
   \]

12. We already know that:
   \[
   \frac{b+c}{a} = m
   \]
   \[
   \frac{a+c}{b} = n
   \]
   \[
   \frac{a+b}{c} = x
   \]

13. Substituting these back into the equation:
   \[
   1 = \frac{2 + m + n + x}{mnx}
   \]

14. Rearrange the equation:
   \[
   mnx = 2 + m + n + x
   \]
   \[
   mnx - x = 2 + m + n
   \]
   \[
   x(mn - 1) = 2 + m + n
   \]
   \[
   x = \frac{2 + m + n}{mn - 1}
   \]

The final answer is \(\boxed{\frac{2 + m + n}{mn - 1}}\)