Question

Cho 3 số a,b,c thỏa mãn abc=2010. Tính giá trị biểu thức:

M= \(\frac{2010a}{ab+2010a+2010}+\frac{b}{bc+b+2010}+\frac{c}{ac+c+1}\)

Answer 1

Ta có: \(M=\frac{2010a}{ab+2010a+2010}+\frac{b}{bc+b+2010}+\frac{c}{ac+c+1}\)

Thế: abc = 2010 ta được:

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

\(\Leftrightarrow\frac{a^2bc}{ab\left(1+ac+c\right)}+\frac{b}{b\left(c+1+ac\right)}+\frac{c}{ac+c+1}\)

\(\Leftrightarrow\frac{a^2bc}{ab\left(1+ac+c\right)}+\frac{ab}{ab\left(c+1+ac\right)}+\frac{abc}{ab\left(ac+c+1\right)}\)

\(\Leftrightarrow\frac{a^2bc+ab+abc}{ab\left(1+ac+c\right)}=\frac{ab\left(ac+1+c\right)}{ab\left(1+ac+c\right)}=1\)

Vậy \(M=1\)