If [math]\frac{1}{a} + \frac{1}{b} + \frac{1}{c} =\frac{1}{a+b+c}[/math] where [math]a+b+c \neq 0, abc \neq 0,[/math] what is the value of [math](a+b) (b+c) (c+a)?[/math] If [math]a, b,[/math] and [math]c[/math] are positive real numbers, and [math]a \lt b + c[/math], how can you prove that [math]\dfrac{a}{1+a} < \dfrac{b}{1+b} + \dfrac{c}{1+c}[/math]?