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\(\frac{a}{b+c+d}=\frac{b}{a+c+d}=\frac{c}{a+b+d}=\frac{d}{a+b+c}\)\(\Rightarrow\frac{a}{b+c+d}+1=\frac{b}{a+c+d}+1=\frac{c}{a+b+d}+1=\frac{d}{a+b+c}+1\)
\(\Rightarrow\frac{a+b+c+d}{b+c+d}=\frac{a+b+c+d}{a+c+d}=\frac{a+b+c+d}{a+b+d}\)\(=\frac{a+b+c+d}{a+b+c}\)
Do a + b + c + d khác 0 nên: b+c+d = a+c+d = a+b+d = a+b+c => a = b = c = d
\(\Rightarrow A=\frac{a+b}{c+d}+\frac{b+c}{a+d}+\frac{c+d}{a+b}+\frac{d+a}{b+c}=\frac{a+a}{a+a}+\frac{b+b}{b+b}+\frac{c+c}{c+c}+\frac{d+d}{d+d}\)\(\left(a=b=c=d\right)\)
\(\Rightarrow A=1+1+1+1=4\)
ta nhân lần lượt a,b,c,d vào biểu thức ban đầu , được
\(\hept{\begin{cases}\frac{a^2}{b+c+d}+\frac{ba}{a+c+d}+\frac{ac}{a+b+d}+\frac{ad}{a+b+c}=a\left(1\right)\\\frac{ab}{b+c+d}+\frac{b^2}{a+c+d}+\frac{cb}{a+b+d}+\frac{db}{a+b+c}=b\left(2\right)\end{cases}}\)
\(\hept{\begin{cases}\frac{ac}{b+c+d}+\frac{bc}{c+a+d}+\frac{c^2}{a+b+d}+\frac{dc}{a+b+c}=c\left(3\right)\\\frac{ad}{b+c+d}+\frac{bd}{a+c+d}+\frac{cd}{a+b+d}+\frac{d^2}{a+b+c}=d\left(4\right)\end{cases}}\)
Lấy (1)+(2)+(3)+(4) ta có :
\(\left(\frac{a^2}{b+c+d}+\frac{b^2}{a+c+d}+\frac{c^2}{a+b+d}+\frac{d^2}{a+b+c}\right)+\frac{ab+bc+bd}{c+d+a}+\frac{ac+bc+cd}{d+a+b}\)
\(+\frac{ad+bd+cd}{a+b+c}+\frac{ab+ac+ad}{b+c+d}=a+b+c+d\)
\(< =>A+\frac{b\left(c+d+a\right)}{c+d+a}+\frac{d\left(a+b+c\right)}{a+b+c}+\frac{c\left(b+d+a\right)}{b+d+a}+\frac{a\left(c+b+d\right)}{c+b+d}=a+b+c+d\)
\(< =>A+a+b+c+d=a+b+c+d=>A=0\)
Vậy \(A=\frac{a^2}{b+c+d}+\frac{b^2}{a+c+d}+\frac{c^2}{a+b+d}+\frac{d^2}{a+b+c}=0\)
