Question

Cho a + b + c = 2007 và 1/a + b + 1/b+c + 1/c + a = 1/90

Tính : S = a/b+c + b/c+a + c/a+b

Answer 1

Giải:

Ta có:

\(\dfrac{1}{a+b}+\dfrac{1}{b+c}+\dfrac{1}{c+a}=\dfrac{1}{90}\)

\(\Leftrightarrow\left(a+b+c\right)\left(\dfrac{1}{a+b}+\dfrac{1}{b+c}+\dfrac{1}{c+a}\right)=\dfrac{a+b+c}{90}\)

\(\Leftrightarrow\dfrac{a+b+c}{a+b}+\dfrac{a+b+c}{b+c}+\dfrac{a+b+c}{c+a}=\dfrac{a+b+c}{90}\)

\(\Leftrightarrow\dfrac{c}{a+b}+1+\dfrac{a}{b+c}+1+\dfrac{b}{c+a}+1=\dfrac{2007}{90}\)

\(\Leftrightarrow\dfrac{c}{a+b}+\dfrac{a}{b+c}+\dfrac{b}{c+a}=\dfrac{2007}{90}-3\)

\(\Leftrightarrow S=19,3\)

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