\(\frac{1}{m}+\frac{1}{n}+\frac{1}{p}=1\Leftrightarrow\)\(\frac{mn+np+mp}{mnp}=1\Leftrightarrow mn+np+mp=mnp\)
Ta có: \(am^3=bn^3=cp^3\Leftrightarrow\)\(\sqrt[3]{am^3}=\sqrt[3]{bn^3}=\sqrt[3]{cp^3}\)\(\Leftrightarrow\sqrt[3]{a}m=\sqrt[3]{b}n=\sqrt[3]{c}p\)
\(\frac{\sqrt[3]{a}m}{mnp}=\frac{\sqrt[3]{b}n}{mnp}=\frac{\sqrt[3]{c}p}{mnp}\Leftrightarrow\)\(\frac{\sqrt[3]{a}}{np}=\frac{\sqrt[3]{b}}{mp}=\frac{\sqrt[3]{c}}{mn}\Leftrightarrow\)\(\frac{\sqrt[3]{a}}{np}=\frac{\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}}{mn+np+mp}\Leftrightarrow\)\(\frac{\sqrt[3]{a}}{np}=\frac{\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}}{mnp}\Leftrightarrow\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}=\sqrt[3]{a}m\)
Mặt khác: \(am^3=bn^3=cp^3\Leftrightarrow\)\(\frac{am^3}{mnp}=\frac{bn^3}{mnp}=\frac{cp^3}{mnp}\Leftrightarrow\)\(\frac{am^2}{np}=\frac{bn^2}{mp}=\frac{cp^2}{mn}\Leftrightarrow\)
\(\frac{am^2}{np}=\frac{am^2+bn^2+cp^2}{mn+np+mp}=\frac{am^2+bn^2+cp^2}{mnp}\)\(\Leftrightarrow am^2+bn^2+cp^2=am^3\Leftrightarrow\sqrt[3]{am^2+bn^2+cp^2}=\sqrt[3]{a}m\)
Vậy =>dpcm