OMK 2019

Given three nonzero real numbers $a,b,c,$ such that $a>b>c$ , prove the equation has at least one real root. $$\frac{1}{x+a}+\frac{1}{x+b}+\frac{1}{x+c}-\frac{3}{x}=0$$ @below sorry, I believe I fixed it with the added constraint.

Given a parallelogram $ABCD$ , a point M is chosen such that $\angle DAC=\angle MAC$ and $\angle CAB=\angle MAB.$ Prove $\frac{AM}{BM}=\left(\frac{AC}{BD}\right)^2$

An arithmetic sequence of five terms is considered $good$ if it contains 19 and 20. For example, $18.5,19.0,19.5,20.0,20.5$ is a $good$ sequence. For every $good$ sequence, the sum of its terms is totalled. What is the total sum of all $good$ sequences?