HMMT November 2014 E8

Let $a, b, c, x$ be reals with $(a+b)(b+c)(c+a) \neq 0$ that satisfy

$$\frac{a^{2}}{a+b}=\frac{a^{2}}{a+c}+20, \frac{b^{2}}{b+c}=\frac{b^{2}}{b+a}+14, \text { and } \frac{c^{2}}{c+a}=\frac{c^{2}}{c+b}+x$$

Compute $x$.