\(\dfrac{xy}{x+y}=\dfrac{yz}{y+z}=\dfrac{zx}{z+x}\\ \Leftrightarrow\dfrac{x+y}{xy}=\dfrac{y+z}{yz}=\dfrac{z+x}{zx}\\ \Leftrightarrow\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{1}{y}+\dfrac{1}{z}=\dfrac{1}{z}+\dfrac{1}{x}\\ \Leftrightarrow\dfrac{1}{x}=\dfrac{1}{y}=\dfrac{1}{z}\Leftrightarrow x=y=z\\ \Leftrightarrow A=\dfrac{2020x^3+2020x^3+2020x^3}{3x^3}+\dfrac{2021x^5+2021x^5}{2x^5}\\ A=\dfrac{6060x^3}{3x^3}+\dfrac{4042x^5}{2x^5}=2020+2021=4041\)
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