Vì $\frac{ab}{a+b} = \frac{bc}{b+c} = \frac{ca}{c+a}$
$\Rightarrow \frac{a+b}{ab} = \frac{b+c}{bc} = \frac{c+a}{ca}$
$\Rightarrow \frac{1}{b}+\frac{1}{a}=\frac{1}{c}+\frac{1}{b}=\frac{1}{a}+\frac{1}{c}$
Ta có:
$\left\{ \begin{align}
& \frac{1}{b}+\frac{1}{a}=\frac{1}{c}+\frac{1}{b} \\
& \frac{1}{c}+\frac{1}{b}=\frac{1}{a}+\frac{1}{c} \\
& \frac{1}{a}+\frac{1}{c}=\frac{1}{b}+\frac{1}{a} \\
\end{align} \right.$$\Rightarrow \frac{1}{a}=\frac{1}{b}=\frac{1}{c}\Rightarrow a=b=c$
Thay vào P, ta có :
$P=2021\text{a}-2020.\frac{a}{a}+2021\text{a}-2020.\frac{a}{a}+2021\text{a}-2020.\frac{a}{a}=3$