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a) \(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}=\frac{1}{5}\)
\(\Leftrightarrow\frac{2015}{a+b}+\frac{2015}{b+c}+\frac{2015}{c+a}=403\)
\(\Leftrightarrow\frac{a+b+c}{a+b}+\frac{a+b+c}{b+c}+\frac{a+b+c}{c+a}=403\)
\(\Leftrightarrow3+\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}=403\)
\(\Leftrightarrow\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}=400\)
\(S=\frac{2015-\left(a+b\right)}{a+b}+\frac{2015-\left(b+c\right)}{b+c}+\frac{2015-\left(a+c\right)}{a+c}=\frac{2015}{a+b}-\frac{a+b}{a+b}+\frac{2015}{b+c}-\frac{b+c}{b+c}+\frac{2015}{a+c}-\frac{a+c}{a+c}\)
\(S=2015.\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{a+c}\right)-3=2015.\frac{1}{10}-3=\frac{1085}{10}\)
Ta có :
\(A+3=\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}+3\)
\(=\left(\frac{a}{b+c}+1\right)+\left(\frac{b}{a+c}+1\right)+\left(\frac{c}{a+b}+1\right)\)
\(=\frac{a+b+c}{b+c}+\frac{a+b+c}{a+c}+\frac{a+b+c}{a+b}\)
\(=\left(a+b+c\right)\left(\frac{1}{b+c}+\frac{1}{a+c}+\frac{1}{a+b}\right)\)
\(=2017.\frac{1}{2017}=1\)
\(\Rightarrow A=1-3=-2\)
\(\left(a+b+c\right)\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)\)
\(=\frac{a+b+c}{a+b}+\frac{a+b+c}{b+c}+\frac{a+b+c}{c+a}\)
\(=1+\frac{c}{a+b}+1+\frac{a}{b+c}+1+\frac{b}{c+a}\)
\(=3+Q\)
Suy ra \(3+Q=1\Leftrightarrow Q=-2\).
\(S=\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\)
\(\Rightarrow S=\left(\frac{a}{b+c}+1\right)+\left(\frac{b}{a+c}+1\right)+\left(\frac{c}{a+b}+1\right)-3\)
\(\Rightarrow S=\frac{a+b+c}{b+c}+\frac{a+b+c}{a+c}+\frac{a+b+c}{a+b}-3\)
\(\Rightarrow S=\left(a+b+c\right).\left(\frac{1}{b+c}+\frac{1}{a+c}+\frac{1}{a+b}\right)-3\)
\(\Rightarrow S=4034.\frac{1}{2}-3=2014\)
Vậy S=2014
\(P=\left(1+\frac{b}{a}\right)\left(1+\frac{a}{c}\right)\left(1+\frac{c}{b}\right)=\frac{a+b}{a}.\frac{a+c}{c}.\frac{b+c}{b}\)
Ta có: \(\frac{a+b-c}{c}=\frac{b+c-a}{a}=\frac{c+a-b}{b}=\frac{a+b}{c}-1=\frac{b+c}{a}-1=\frac{c+a}{b}-1\)
\(\Rightarrow\frac{a+b}{c}=\frac{b+c}{a}=\frac{c+a}{b}=\frac{a+b+b+c+c+a}{c+a+b}=\frac{2\left(a+b+c\right)}{a+b+c}\)
TH1: \(a+b+c=0\)\(\Rightarrow\hept{\begin{cases}a+b=-c\\b+c=-a\\c+a=-b\end{cases}}\)
\(\Rightarrow P=\frac{-c}{a}.\frac{-b}{c}.\frac{-a}{b}=\frac{\left(-a\right).\left(-b\right).\left(-c\right)}{a.b.c}=-1\)
TH2: \(a+b+c\ne0\)\(\Rightarrow\frac{a+b}{c}=\frac{b+c}{a}=\frac{c+a}{b}=2\)
\(\Rightarrow\hept{\begin{cases}a+b=2c\\b+c=2a\\c+a=2b\end{cases}}\)\(\Rightarrow P=\frac{2c}{a}.\frac{2b}{c}.\frac{2a}{b}=\frac{8abc}{abc}=8\)
Vậy \(P=-1\)hoặc \(P=8\)
sửa đề câu 1.
cho \(\frac{a}{b+c}=\frac{b}{a+c}=\frac{c}{a+b}\)
...
giải
cộng 1 vào mỗi tỉ số ta được :
\(\frac{a}{b+c}+1=\frac{b}{a+c}+1=\frac{c}{a+b}+1\)
hay \(\frac{a+b+c}{b+c}=\frac{a+b+c}{a+c}=\frac{a+b+c}{a+b}\)
+) nếu a + b + c = 0 thì :
b + c = -a ; a + c = -b ; a + b = -c
\(\Rightarrow P=\frac{a}{-a}+\frac{b}{-b}+\frac{c}{-c}=-1+\left(-1\right)+\left(-1\right)=-3\)
+ ) nếu a + b + c \(\ne\)0 thì : a = b = c
\(\Rightarrow P=\frac{1}{2}+\frac{1}{2}+\frac{1}{2}=\frac{3}{2}\)
Vậy ...
2) Áp dụng tính chất của dãy tỉ số bằng nhau, ta có :
\(\frac{a}{2014}=\frac{b}{2015}=\frac{c}{2016}=\frac{a-b}{2014-2015}=\frac{b-c}{2015-2016}=\frac{c-a}{2016-2014}\)
hay \(\frac{a-b}{-1}=\frac{b-c}{-1}=\frac{c-a}{2}\)
\(\Rightarrow\left(\frac{a-b}{-1}\right).\left(\frac{b-c}{-1}\right)=\left(\frac{c-a}{2}\right)^2\)
\(\Rightarrow\left(a-b\right).\left(b-c\right)=\frac{\left(c-a\right)^2}{4}\)
\(\Rightarrow4.\left(a-b\right).\left(b-c\right)=\left(c-a\right)^2\)
Vậy ...
Cộng 1 vào mỗi phân số của B rồi trừ 3 đi là đd