\(\orbr{\begin{cases}\\\end{cases}}\)

\(\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}=2\Leftrightarrow1-\frac{a}{a+b}-\frac{b}{b+c}+1-\frac{c}{c+d}-\frac{d}{d+a}=0\)

\(\Leftrightarrow\frac{b\left(c-a\right)}{\left(a+b\right)\left(b+c\right)}+\frac{d\left(a-c\right)}{\left(c+d\right)\left(d+a\right)}=0\)

\(\Leftrightarrow b\left(c-a\right)\left(a+b\right)\left(b+c\right)-d\left(c-a\right)\left(c+d\right)\left(d+a\right)=0\)

\(\Leftrightarrow b\left(a+b\right)\left(b+c\right)-d\left(c+d\right)\left(d+a\right)=0\)

\(\Leftrightarrow bad+bd^2+bca+bcd-dab-dac-db^2-cbd=0\)

\(\Leftrightarrow bca-dca+bd^2-db^2=0\)

\(\Leftrightarrow\left(b-d\right)\left(ca-bd\right)=0\)

\(\Rightarrow ca=bd\Rightarrow abcd=bd^2\)

sao cái dấu tương đương thứ 4 bạn bỏ c-a v ạ

Tách ra bạn có: \(\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}=2\Leftrightarrow1-\frac{a}{a+b}-\frac{b}{b+c}+1-\frac{c}{c+d}-\frac{d}{d+a}=0\)

Quy đồng: \(\Leftrightarrow\frac{b\left(c-a\right)}{\left(a+b\right)\left(b+c\right)}+\frac{d\left(a-c\right)}{\left(c+d\right)\left(d+a\right)}=0\)

\(\Leftrightarrow b\left(c-a\right)\left(a+b\right)\left(b+c\right)-d\left(c-a\right)\left(c+d\right)\left(d+a\right)=0\)

Do a<>c:

\(\Leftrightarrow b\left(a+b\right)\left(b+c\right)-d\left(c+d\right)\left(d+a\right)=0\)

Phá ngoặc:

\(\Leftrightarrow bad+bd^2+bca+bcd-dab-dac-db^2-cbd=0\)

\(\Leftrightarrow bca-dca+bd^2-db^2=0\)

Phân tích đa thức thành nhân tử:

\(\Leftrightarrow\left(b-d\right)\left(ca-bd\right)=0\)

Do b<>d:

\(\Rightarrow ca=bd\Rightarrow abcd=bd^2\)

Câu hỏi của Trần Anh Đại  nếu ko vào được ib vs tui  để biết thêm chi tiết!

Câu hỏi của Trần Anh Đại:bạn tham khảo tại đây!

Tách ra bạn có: \(\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}=2\Leftrightarrow1-\frac{a}{a+b}-\frac{b}{b+c}+1-\frac{c}{c+d}-\frac{d}{d+a}=0\)

Quy đồng: \(\Leftrightarrow\frac{b\left(c-a\right)}{\left(a+b\right)\left(b+c\right)}+\frac{d\left(a-c\right)}{\left(c+d\right)\left(d+a\right)}=0\)

\(\Leftrightarrow b\left(c-a\right)\left(a+b\right)\left(b+c\right)-d\left(c-a\right)\left(c+d\right)\left(d+a\right)=0\)

Do a<>c:

\(\Leftrightarrow b\left(a+b\right)\left(b+c\right)-d\left(c+d\right)\left(d+a\right)=0\)

Phá ngoặc:

\(\Leftrightarrow bad+bd^2+bca+bcd-dab-dac-db^2-cbd=0\)

\(\Leftrightarrow bca-dca+bd^2-db^2=0\)

Phân tích đa thức thành nhân tử:

\(\Leftrightarrow\left(b-d\right)\left(ca-bd\right)=0\)

Do b<>d:

\(\Rightarrow ca=bd\Rightarrow abcd=bd^2\)

Thỏa mãn.

Mình không chắc câu này lắm nhưng thôi giải dùm bạn vậy :((

\(\frac{2a+b}{a+b}+\frac{2b+c}{b+c}+\frac{2c+d}{c+d}+\frac{2d+a}{d+a}=6\)

\(\Leftrightarrow\)\(1+\frac{a}{a+b}+1+\frac{b}{b+c}+1+\frac{c}{c+d}+1+\frac{d}{d+a}=6\)

\(\Leftrightarrow\)\(\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}=2\)

\(\Leftrightarrow\)\(1-\frac{a}{a+b}-\frac{b}{b+c}+1-\frac{c}{c+d}-\frac{d}{d+a}=0\)

\(\Leftrightarrow\)\(\frac{b}{a+b}-\frac{b}{b+c}+\frac{d}{c+d}-\frac{d}{d+a}=0\)

\(\Leftrightarrow\)\(\frac{b\left(c-a\right)}{\left(a+b\right)\left(b+c\right)}+\frac{d\left(a-c\right)}{\left(c+d\right)\left(d+a\right)}=0\)

\(\Leftrightarrow\)\(b\left(c+d\right)\left(d+a\right)-d\left(a+b\right)\left(b+c\right)\)

\(\Leftrightarrow\)\(abc-acd+bd^2-b^2d=0\)

\(\Leftrightarrow\)\(\left(b-d\right)\left(ac-bd\right)=0\)

\(\Leftrightarrow\)\(ac-bd=0\Leftrightarrow ac=bd\left(b\ne d\right)\)

Vậy bạn tự kết luận nha

\(\Leftrightarrow1+\frac{a}{a+b}+1+\frac{b}{b+c}+1+\frac{c}{c+d}+1+\frac{d}{d+a}=6\)

\(\Leftrightarrow\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}+\frac{d}{d+a}=2\)

\(\Leftrightarrow1-\frac{a}{a+b}-\frac{b}{b+c}+1-\frac{c}{c+d}-\frac{d}{d+a}=0\)

\(\Leftrightarrow\frac{b}{a+b}-\frac{b}{b+c}+\frac{d}{c+d}-\frac{d}{d+a}=0\)

\(\Leftrightarrow\frac{b\left(b+c\right)-b\left(a+b\right)}{\left(a+b\right)\left(b+c\right)}+\frac{d\left(d+a\right)-d\left(c+d\right)}{\left(c+d\right)\left(d+a\right)}=0\)

\(\Leftrightarrow\frac{b\left(c-a\right)}{\left(a+b\right)\left(b+c\right)}+\frac{d\left(a-c\right)}{\left(c+d\right)\left(d+a\right)}=0\)

\(\Leftrightarrow b\left(c-a\right)\left(c+d\right)\left(d+a\right)+d\left(a-c\right)\left(a+b\right)\left(b+c\right)=0\)

\(\Leftrightarrow b\left(c-a\right)\left(c+d\right)\left(d+a\right)-d\left(c-a\right)\left(a+b\right)\left(b+c\right)=0\)

\(\Leftrightarrow b\left(c+d\right)\left(d+a\right)-d\left(a+b\right)\left(b+c\right)=0\)

\(\Leftrightarrow\left(bc+bd\right)\left(d+a\right)-\left(da+db\right)\left(b+c\right)=0\)

\(\Leftrightarrow bcd+bca+bd^2+bda-abd-adc-db^2-dbc=0\)

\(\Leftrightarrow bca-acd+bd^2-b^2d=0\)

\(\Leftrightarrow ac\left(b-d\right)-bd\left(b-d\right)=0\)

\(\Leftrightarrow\left(b-d\right)\left(ac-bd\right)=0\)

\(\Leftrightarrow ac-bd=0\)

\(\Leftrightarrow ac=bd\)

\(\Leftrightarrow\left(ac\right)^2=abcd\)\(\left(đpcm\right)\)

dành cho người không hiểu bài trên 

                                                                           \(#huybip#\)

Điều kiện đã cho có thể được viết lại thành \(\dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{c+d}+\dfrac{d}{d+a}=2\)

hay \(1-\dfrac{a}{a+b}-\dfrac{b}{b+c}+1-\dfrac{c}{c+d}-\dfrac{d}{d+a}=0\)

\(\Leftrightarrow\dfrac{b}{a+b}-\dfrac{b}{b+c}+\dfrac{d}{c+d}-\dfrac{d}{d+a}=0\)

\(\Leftrightarrow\dfrac{b^2+bc-ab-b^2}{\left(a+b\right)\left(b+c\right)}+\dfrac{d^2+da-cd-d^2}{\left(c+d\right)\left(d+a\right)}=0\)

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

\(\Leftrightarrow\left(c-a\right)\left[\dfrac{b}{\left(a+b\right)\left(b+c\right)}-\dfrac{d}{\left(c+d\right)\left(d+a\right)}\right]=0\)

\(\Leftrightarrow\dfrac{b}{\left(a+b\right)\left(b+c\right)}=\dfrac{d}{\left(c+d\right)\left(d+a\right)}\) (do \(c\ne a\))

\(\Leftrightarrow b\left(cd+ca+d^2+da\right)=d\left(ab+ac+b^2+bc\right)\)

\(\Leftrightarrow bcd+abc+bd^2+abd=abd+acd+b^2d+bcd\)

\(\Leftrightarrow abc+bd^2-acd-b^2d=0\)

\(\Leftrightarrow ac\left(b-d\right)-bd\left(b-d\right)=0\)

\(\Leftrightarrow\left(b-d\right)\left(ac-bd\right)=0\)

\(\Leftrightarrow ac=bd\) (do \(b\ne d\))

 Do đó \(A=abcd=ac.ac=\left(ac\right)^2\), mà \(a,c\inℕ^∗\) nên A là SCP (đpcm)

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