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TH1: \(a+b+c+d\ne0\)
\(\dfrac{2a+b+c+d}{a}=\dfrac{a+2b+c+d}{b}=\dfrac{a+b+2c+d}{c}=\dfrac{a+b+c+2d}{d}\)
\(\Rightarrow\dfrac{2a+b+c+d}{a}-1=\dfrac{a+2b+c+d}{b}-1=\dfrac{a+b+2c+d}{c}-1=\dfrac{a+b+c+2d}{d}-1\)
\(\Rightarrow\dfrac{a+b+c+d}{a}=\dfrac{a+b+c+d}{b}=\dfrac{a+b+c+d}{c}=\dfrac{a+b+c+2d}{d}\)
\(\Rightarrow a=b=c=d\)
\(P=\dfrac{a+b}{c+d}+\dfrac{b+c}{d+a}+\dfrac{c+d}{a+b}+\dfrac{d+a}{b+c}\)
\(\Rightarrow P=1+1+1+1\)
\(\Rightarrow P=4\)
TH2: \(a+b+c+d=0\)
\(\Rightarrow\left\{{}\begin{matrix}a+b=-\left(c+d\right)\\b+c=-\left(d+a\right)\\c+d=-\left(a+b\right)\\d+a=-\left(b+c\right)\end{matrix}\right.\)
\(P=\dfrac{a+b}{c+d}+\dfrac{b+c}{d+a}+\dfrac{c+d}{a+b}+\dfrac{d+a}{b+c}\)
\(\Rightarrow P=\dfrac{-\left(c+d\right)}{c+d}+\dfrac{-\left(d+a\right)}{d+a}+\dfrac{-\left(a+b\right)}{a+b}+\dfrac{-\left(b+c\right)}{b+c}\)
\(\Rightarrow P=-1+\left(-1\right)+\left(-1\right)+\left(-1\right)\)
\(\Rightarrow P=-4\)
bn mình nền của bn là nôb team trưởng team là t gaming
\(\dfrac{a}{b+c+d}=\dfrac{b}{a+c+d}=\dfrac{c}{a+b+d}=\dfrac{d}{a+b+c}=\dfrac{a+b+c+d}{3\left(a+b+c+d\right)}=\dfrac{1}{3}\\ \Rightarrow\left\{{}\begin{matrix}b+c+d=3a\\a+c+d=3b\\a+b+d=3c\\a+b+c=3d\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}a+b+c+d=2a\\a+b+c+d=2b\\a+b+c+d=2c\\a+b+c+d=2d\end{matrix}\right.\\ \Rightarrow2a=2b=2c=2d\\ \Rightarrow a=b=c=d\\ \Rightarrow A=\dfrac{a+a}{a+a}+\dfrac{a+a}{a+a}+\dfrac{a+a}{a+a}+\dfrac{a+a}{a+a}=1+1+1+1=4\)
Ta có:
\(\dfrac{2a+b+c+d}{a}=\dfrac{a+2b+c+d}{b}=\dfrac{a+b+2c+d}{c}=\dfrac{a+b+c+2d}{d}\)
⇔ \(\dfrac{2a+b+c+d}{a}-1=\dfrac{a+2b+c+d}{b}-1=\dfrac{a+b+2c+d}{c}-1\)
\(=\dfrac{a+b+c+2d}{d}-1\)
⇔ \(\dfrac{a+b+c+d}{a}=\dfrac{a+b+c+d}{b}=\dfrac{a+b+c+d}{c}=\dfrac{a+b+c+d}{d}\)
Nếu a+b+c+d=0
⇒a+b=−(c+d);c+b=−(a+d);c+d=−(a+b);a+d=−(c+b)
Thay vào M, ta có:
\(M=\dfrac{a+b}{-\left(a+b\right)}=\dfrac{b+c}{-\left(b+c\right)}=\dfrac{c+d}{-\left(c+d\right)}=\dfrac{a+d}{-\left(a+d\right)}=-1\)
Nếu a+b+c+d ≠0
⇒ \(a=b=c=d\)
Thay vào M, ta có
\(M=\dfrac{a+b}{a+b}=\dfrac{b+c}{b+c}=\dfrac{c+d}{c+d}=\dfrac{d+a}{d+a}=1\)
\(TH1:a+b+c+d\ne0\)
\(\dfrac{2a+b+c+d}{a}=\dfrac{a+2b+c+d}{b}=\dfrac{a+b+2c+d}{c}=\dfrac{a+b+c+2d}{d}\)
\(\Rightarrow\dfrac{2a+b+c+d}{a}-1=\dfrac{a+2b+c+d}{b}-1=\dfrac{a+b+2c+d}{c}-1=\dfrac{a+b+c+2d}{d}-1\)
\(\Rightarrow\dfrac{a+b+c+d}{a}=\dfrac{a+b+c+d}{b}=\dfrac{a+b+c+d}{c}=\dfrac{a+b+c+d}{d}\)
\(\Rightarrow a=b=c=d\)
\(M=\dfrac{a+b}{c+d}+\dfrac{b+c}{d+a}+\dfrac{c+d}{a+b}+\dfrac{a+d}{b+c}\)
\(=1+1+1+1\)
\(=4\)
\(TH2:a+b+c+d=0\)
\(\Rightarrow\left\{{}\begin{matrix}a+b=-\left(c+d\right)\\b+c=-\left(d+a\right)\\c+d=-\left(a+b\right)\\d+a=-\left(b+c\right)\end{matrix}\right.\)
\(M=\dfrac{a+b}{c+d}+\dfrac{b+c}{d+a}+\dfrac{c+d}{a+b}+\dfrac{a+d}{b+c}\)
\(=-\dfrac{c+d}{c+d}-\dfrac{d+a}{d+a}-\dfrac{a+b}{a+b}-\dfrac{b+c}{b+c}\)
\(=-1-1-1-1\)
\(=-4\)
TH1:
\(\dfrac{a}{b+c+d}=\dfrac{b}{c+d+a}=\dfrac{c}{d+a+b}=\dfrac{d}{a+b+c}=\dfrac{a+b+c+d}{3\left(a+b+c+d\right)}=\dfrac{1}{3}\)
\(\Rightarrow\left\{{}\begin{matrix}3a=b+c+d\\3b=a+c+d\\3c=a+b+d\\3d=a+b+c\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}3\left(a-b\right)=b-a\\3\left(b-c\right)=c-b\\3\left(c-d\right)=d-c\\3\left(d-a\right)=a-d\end{matrix}\right.\) \(\Rightarrow a=b=c=d\)
\(\Rightarrow P=\dfrac{2a}{2a}+\dfrac{2a}{2a}+\dfrac{2a}{2a}+\dfrac{2a}{2a}=1+1+1+1=4\)
TH2: \(\dfrac{a}{b+c+d}=\dfrac{b}{a+c+d}=\dfrac{c}{a+b+d}=\dfrac{d}{a+b+c}=\dfrac{a-b}{a-a}=-1\)
\(\Rightarrow-a=b+c+d\Rightarrow a+b+c+d=0\)
\(\Rightarrow\left\{{}\begin{matrix}a+b=-\left(c+d\right)\\a+c=-\left(b+d\right)\\a+d=-\left(b+c\right)\end{matrix}\right.\)
\(\Rightarrow P=\dfrac{-\left(c+d\right)}{c+d}+\dfrac{b+c}{-\left(b+c\right)}+\dfrac{c+d}{-\left(c+d\right)}+\dfrac{-\left(b+c\right)}{b+c}=-1+-1+-1+-1=-4\)
Vậy \(\left[{}\begin{matrix}P=4\\P=-4\end{matrix}\right.\)
:)
- Ta có: \(\dfrac{a}{b}< \dfrac{c}{d}\) (gt)
=>\(ad< bc\)
=>\(ad+ab< bc+ab\)
=>\(a\left(b+d\right)< b\left(a+c\right)\)
=>\(\dfrac{a}{b}< \dfrac{a+c}{b+d}\) (1)
- Ta có: \(\dfrac{c}{d}>\dfrac{a}{b}\) (gt)
=>\(bc>ad\)
=>\(bc+cd>ad+cd\)
=>\(c\left(b+d\right)>d\left(a+c\right)\)
=>\(\dfrac{c}{d}>\dfrac{a+c}{b+d}\) (2)
- Từ (1) và (2) suy ra: \(\dfrac{a}{b}< \dfrac{a+c}{b+d}< \dfrac{c}{d}\)
\(\dfrac{a}{b+c+d}=\dfrac{b}{c+a+d}=\dfrac{c}{a+b+d}=\dfrac{d}{a+b+c}\)
+)Xét a+b+c+d=0 thì a+d=-c-d
b+c=-d-a
c+d=-b-a
d+a=-b-c
Do đó:
\(P=\dfrac{-c-d}{c+d}+\dfrac{-a-b}{a+b}+\dfrac{-b-c}{b+c}+\dfrac{-d-a}{a+d}\\ =-1+-1+-1+-1=-4\)
+)Xét a+b+c+d khác 0
áp dụng t/c của dãy tỉ số bằng nhau, ta có:
\(\dfrac{a}{b+c+d}=\dfrac{b}{a+c+d}=\dfrac{c}{a+b+d}=\dfrac{d}{a+c+d}=\dfrac{a+b+c+d}{3\left(a+b+c+d\right)}=\dfrac{1}{3}\)
=>\(a=\dfrac{1}{3}\left(d+b+c\right)\)
\(b=\dfrac{1}{3}\left(a+c+d\right)\)
\(c=\dfrac{1}{3}\left(a+b+d\right)\)
\(d=\dfrac{1}{3}\left(a+b+c\right)\)
Bạn thay vào r tính
Ta có : \(\dfrac{a}{b+c+d}=\dfrac{b}{c+d+a}=\dfrac{c}{d+a+b}=\dfrac{d}{a+b+c}\)
\(\Rightarrow\)\(\dfrac{a}{b+c+d}+1=\dfrac{b}{c+d+a}+1=\dfrac{c}{d+a+b}+1=\dfrac{d}{a+b+c}+1\)
\(\Rightarrow\)\(\dfrac{a+b+c+d}{b+c+d}=\dfrac{b+c+d+a}{c+d+a}=\dfrac{c+d+a+b}{d+a+b}=\dfrac{d+a+b+c}{a+b+c}\)
TH1 : \(a+b+c+d\ne0\)\(\Rightarrow\) \(a=b=c=d\)
\(\Rightarrow\) P= \(\dfrac{a+b}{c+d}+\dfrac{b+c}{d+a}+\dfrac{c+d}{b+a}+\dfrac{d+a}{b+c}=1+1+1+1=4\)
TH2 : \(a+b+c+d=0\Rightarrow\left\{{}\begin{matrix}a+b=-\left(c+d\right)\\b+c=-\left(a+d\right)\end{matrix}\right.\)
\(\Rightarrow P=\dfrac{a+b}{c+d}+\dfrac{b+c}{d+a}+\dfrac{c+d}{b+a}+\dfrac{d+a}{b+c}=-1+\left(-1\right)+\left(-1\right)+\left(-1\right)=-4\)