Problem on function + simple inequality from AL discussion.

Recently solved a problem that is given from my classmate. (the solution is confirmed to be right by Kin Li, at least at the moment I post the question here) You can try the following one.

Problem 1. Let $f:[0,1]\to[0,1],g:[0,1]\to[0,1]$ be continuous and satisfy $f\circ g(x)=g\circ f(x)$. Prove that there is a $w\in [0,1]$ such that $f(w)=g(w)$.

I think it is not that easy. :)

Problem 2. Suppose $a,b,c >0$ and $ab+bc+ca=\frac{1}{3}$, show that $$\frac{a}{a^2-bc+1}+\frac{b}{b^2-ca+1}+\frac{c}{c^2-ab+1}\ge \frac{1}{a+b+c}.$$

Once again, AL knowledge is enough.