Let the sum of all possible values of $x$ be $p/q$, where $p,q$ are relatively prime positive integers. Compute $p+q$.
This problem is somewhat perplexing. Letting $a=x_0$, I got that $$f(a)=3a^5+3a^2+a-4=0.$$ Now by graphing, it is clear that this quintic only has 1 real solution (alternatively, you could take the derivative and notice how sharp it is). The root seems to be around 0.83, and the rational root test fails. At this point I conclude that the problem had a mistake. Given that this problem was on a recent competition, this seems unlikely.
Is there any interpretation of the problem, or any notion of an intended interpretation that would lead to an existing solution?