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\(\frac{ab}{a+b}=\frac{bc}{b+c}=\frac{ac}{c+a}\Leftrightarrow\frac{a+b}{ab}=\frac{b+c}{bc}=\frac{c+a}{ac}\)
\(\Leftrightarrow\frac{1}{a}+\frac{1}{b}=\frac{1}{b}+\frac{1}{c}=\frac{1}{c}+\frac{1}{a}\)
\(\Leftrightarrow\hept{\begin{cases}\frac{1}{a}+\frac{1}{b}=\frac{1}{b}+\frac{1}{c}\\\frac{1}{b}+\frac{1}{c}=\frac{1}{c}+\frac{1}{a}\\\frac{1}{c}+\frac{1}{a}=\frac{1}{a}+\frac{1}{b}\end{cases}}\)
\(\Leftrightarrow\frac{1}{a}=\frac{1}{b}=\frac{1}{c}\Leftrightarrow a=b=c\)
Thay vào M được \(M=\frac{3a^2}{3a^2}=1\)
\(\hept{\begin{cases}\frac{ab}{a+b}=\frac{bc}{b+c}\Rightarrow ab.\left(b+c\right)=\left(a+b\right).bc=ab^2+abc=abc+b^2c\\\frac{bc}{b+c}=\frac{ca}{c+a}\Rightarrow\left(a+c\right).bc=\left(b+c\right).ac\Rightarrow abc=c^2a=abc+c^2b\end{cases}}\)
\(\Rightarrow\hept{\begin{cases}a=c\\a=b\end{cases}\Rightarrow a=b=c\Rightarrow M=\frac{ab+bc+ca}{a^2+b^2+c^2}=\frac{a^2+b^2+c^2}{a^2+b^2+c^2}=1}\)
Từ \(\frac{ab}{a+b}=\frac{bc}{b+c}=\frac{ca}{c+a}\)
\(\Rightarrow\frac{a+b}{ab}=\frac{b+c}{bc}=\frac{c+a}{ca}\)
\(\Rightarrow\frac{a}{ab}+\frac{b}{ab}=\frac{b}{bc}+\frac{c}{bc}=\frac{c}{ca}+\frac{a}{ca}\)
\(\Rightarrow\frac{1}{b}+\frac{1}{a}=\frac{1}{c}+\frac{1}{b}=\frac{1}{a}+\frac{1}{c}\)
\(\Rightarrow\frac{1}{a}=\frac{1}{b}=\frac{1}{c}\Rightarrow a=b=c\)
\(\Rightarrow M=\frac{ab+bc+ca}{a^2+b^2+c^2}=\frac{a\cdot a+a\cdot a+a\cdot a}{a^2+a^2+a^2}=\frac{a^2+a^2+a^2}{a^2+a^2+a^2}=1\)
\(\frac{ab}{a+b}=\frac{bc}{b+c}=\frac{ca}{c+a}\) (1)
\(\Rightarrow\frac{a+b}{ab}=\frac{b+c}{bc}=\frac{c+a}{ca}\)
\(\Leftrightarrow\frac{ac+bc}{abc}=\frac{ab+ac}{abc}=\frac{ab+bc}{abc}\)
\(\Rightarrow ac+bc=ab+ac=ab+bc\)
\(\Rightarrow ab=ac=bc\) (2)
Từ (1) và (2)
\(\Rightarrow a=b=c\)
\(\Rightarrow M=\frac{ab+bc+ca}{a^2+b^2+c^2}=\frac{3a^2}{3a^2}=1\)
Vậy M = 1