Question

Câu 1 : Cho các số a,b,c,d thỏa mãn abcd = 1 . Tính giá trị của biểu thức

\(M=\dfrac{1}{abc+ab+a+1}+\dfrac{1}{bcd+bc+b+1}+\dfrac{1}{acb+cd+c+1}+\dfrac{1}{abd+ad+d+1}\)

Answer 1

ta có : \(M=\dfrac{1}{abc+ab+a+1}+\dfrac{1}{bcd+bc+b+1}+\dfrac{1}{acb+cd+c+1}+\dfrac{1}{abd+ad+d+1}\)

\(\Leftrightarrow M=\dfrac{abcd}{abcd+abc+ab+a}+\dfrac{1}{bcd+bc+b+1}+\dfrac{1}{acb+cd+c+1}+\dfrac{1}{abd+ad+d+1}\) \(\Leftrightarrow M=\dfrac{bcd}{bcd+bc+b+1}+\dfrac{1}{bcd+bc+b+1}+\dfrac{1}{acb+cd+c+1}+\dfrac{1}{abd+ad+d+1}\) \(\Leftrightarrow M=\dfrac{bcd+1}{bcd+bc+b+1}+\dfrac{1}{acb+cd+c+1}+\dfrac{1}{abd+ad+d+1}\) \(\Leftrightarrow M=\dfrac{abcd+bcd}{abcd+bcd+bc+b}+\dfrac{1}{acb+cd+c+1}+\dfrac{1}{abd+ad+d+1}\) \(\Leftrightarrow M=\dfrac{acd+cd}{acd+cd+c+1}+\dfrac{1}{acb+cd+c+1}+\dfrac{1}{abd+ad+d+1}\) \(\Leftrightarrow M=\dfrac{acd+cd+1}{acd+cd+c+1}+\dfrac{1}{abd+ad+d+1}\) \(\Leftrightarrow M=\dfrac{abcd+acd+cd}{abcd+acd+cd+c}+\dfrac{1}{abd+ad+d+1}\) \(\Leftrightarrow M=\dfrac{abd+ad+d}{abd+ad+d+1}+\dfrac{1}{abd+ad+d+1}\) \(\Leftrightarrow M=\dfrac{abd+ad+d+1}{abd+ad+d+1}=1\)