Question

[Evil]

1.Cho ab/b = bc/c=ca/a. Tính A= (a-b)(b-c)(c-a) + 2016

2. Cho (ab + bc)/ ( a+b) = ( bc + ca )/(b+c)= ( ca + ab) / (c+a)

Tính M=\(\left(\frac{b}{a}+1\right)\left(\frac{c}{b}+1\right)\left(\frac{a}{c}+1\right)+2016\)

3. Cho a+b+c+d khác 0 và \(\frac{a}{b+c+d}=\frac{b}{a+c+d}=\frac{c}{a+b+d}=\frac{d}{a+b+c}\)

Tìm giá trị của A=\(\frac{a+b}{c+d}+\frac{b+c}{a+d}+\frac{c+d}{a+b}+\frac{d+a}{b+c}\)

Answer 1

[Phùng Minh Quân]

\(1)\)\(\frac{\overline{ab}}{b}=\frac{\overline{bc}}{c}=\frac{\overline{ca}}{a}\)

\(\Leftrightarrow\)\(\frac{10a+b}{b}=\frac{10b+c}{c}=\frac{10c+a}{a}\)

\(\Leftrightarrow\)\(\frac{10a}{b}=\frac{10b}{c}=\frac{10c}{a}\)

Áp dụng tính chất dãy tỉ số bằng nhau ta có : 

\(\frac{10a}{b}=\frac{10b}{c}=\frac{10c}{a}=\frac{10a+10b+10c}{a+b+c}=\frac{10\left(a+b+c\right)}{a+b+c}=10\)

Do đó : 

\(\frac{10a}{b}=10\)\(\Leftrightarrow\)\(a=b\)

\(\frac{10b}{c}=10\)\(\Leftrightarrow\)\(b=c\)

\(\frac{10c}{a}=10\)\(\Leftrightarrow\)\(c=a\)

\(\Rightarrow\)\(a=b=c\)

\(\Rightarrow\)\(A=\left(a-b\right)\left(b-c\right)\left(c-a\right)+2016=2016\)

\(2)\)\(\frac{\overline{ab}+\overline{bc}}{a+b}=\frac{\overline{bc}+\overline{ca}}{b+c}=\frac{\overline{ca}+\overline{ab}}{c+a}\)

Áp dụng tính chất dãy tỉ số bằng nhau ta có : 

\(\frac{\overline{ab}+\overline{bc}}{a+b}=\frac{\overline{bc}+\overline{ca}}{b+c}=\frac{\overline{ca}+\overline{ab}}{c+a}=\frac{2\left(\overline{ab}+\overline{bc}+\overline{ca}\right)}{2\left(a+b+c\right)}=\frac{\overline{ab}+\overline{bc}+\overline{ca}}{a+b+c}\)

\(=\frac{10a+b+10b+c+10c+a}{a+b+c}=\frac{11a+11b+11c}{a+b+c}=\frac{11\left(a+b+c\right)}{a+b+c}=11\)

Do đó : 

\(\frac{\overline{ab}+\overline{bc}}{a+b}=11\)\(\Leftrightarrow\)\(10a+11b+c=11a+11b\)\(\Leftrightarrow\)\(c=a\)

\(\frac{\overline{bc}+\overline{ca}}{b+c}=11\)\(\Leftrightarrow\)\(10b+11c+a=11b+11c\)\(\Leftrightarrow\)\(a=b\)

\(\frac{\overline{ca}+\overline{ab}}{c+a}=11\)\(\Leftrightarrow\)\(10c+11a+b=11c+11a\)\(\Leftrightarrow\)\(b=c\)

\(\Rightarrow\)\(a=b=c\)

\(\Rightarrow\)\(M=\left(\frac{b}{a}+1\right)\left(\frac{c}{b}+1\right)\left(\frac{a}{c}+1\right)+2016=2.2.2+2016=2024\)

Chúc bạn học tốt ~ 

Answer 2

[tth_new]

Ta có: \(\frac{a}{b+c+d}=\frac{b}{a+c+d}=\frac{c}{a+b+d}=\frac{d}{a+b+c}\)

\(\Rightarrow\frac{a}{b+c+d}+1=\frac{b}{a+c+d}+1=\frac{c}{a+b+d}+1=\frac{d}{a+b+c}+1\)

hay \(\frac{a+b+c+d}{b+c+d}=\frac{a+b+c+d}{a+c+d}=\frac{a+b+c+d}{a+b+d}=\frac{a+b+c+d}{a+b+c}\)

Do các tử số trên bằng nhau nên các mẫu số cũng bằng nhau hay \(b+c+d=a+c+d=a+b+d=a+b+c\)

Suy ra a = b =c =d

\(\Rightarrow A=\frac{a+b}{c+d}+\frac{b+c}{a+d}+\frac{c+d}{a+b}+\frac{d+a}{b+c}=1+1+1+1=4\)