Question

cho 3 số a,b,c thỏa mãn

\(\dfrac{bc}{a}+\dfrac{ca}{b}+\dfrac{ab}{c}=a+b+c\)

tính giá trị biểu thức A=\(\dfrac{a^2+b^2}{\left(a+c\right)\left(b+c\right)}+\dfrac{b^2+c^2}{\left(b+a\right)\left(c+a\right)}+\dfrac{c^2+a^2}{\left(c+b\right)\left(a+b\right)}\)

Answer 1

\(\dfrac{bc}{a}+\dfrac{ca}{b}+\dfrac{ab}{c}=a+b+c\)

\(\Leftrightarrow\dfrac{abc}{a^2}+\dfrac{abc}{b^2}+\dfrac{abc}{c^2}=a+b+c\)

\(\Leftrightarrow\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}=\dfrac{a+b+c}{abc}=\dfrac{1}{ab}+\dfrac{1}{bc}+\dfrac{1}{ac}\)

\(\Leftrightarrow\left(\dfrac{1}{a}-\dfrac{1}{b}\right)^2+\left(\dfrac{1}{b}-\dfrac{1}{c}\right)^2+\left(\dfrac{1}{c}-\dfrac{1}{a}\right)^2=0\)

\(\Leftrightarrow a=b=c\)

Thay vào A r tính thôi