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![](https://rs.olm.vn/images/avt/0.png?1311)
+) Do a + b + c> a + b \(\Rightarrow\frac{a}{a+b}>\frac{a}{a+b+c}\)
Tương tự \(\frac{b}{b+c}>\frac{b}{a+b+c},\frac{c}{c+a}>\frac{c}{a+b+c}\)
\(\Rightarrow M>\frac{a}{a+b+c}+\frac{b}{a+b+c}+\frac{c}{a+b+c}=\frac{a+b+c}{a+b+c}=1\)(1)
Lại có a < a + b \(\Rightarrow\frac{a}{a+b}< 1\Rightarrow\frac{a+c}{a+b+c}>\frac{a}{a+b}\)
Tương tự \(\frac{b}{b+c}< \frac{b+a}{a+b+c},\frac{c}{c+a}< \frac{c+b}{a+b+c}\)
\(\Rightarrow M< \frac{a+c}{a+b+c}+\frac{b+a}{a+b+c}+\frac{c+b}{a+b+c}=\frac{2\left(a+b+c\right)}{a+b+c}=2\)(2)
Từ (1) và (2) => 1<M<2 => M không phải là số nguyên
Vì a,b,c dương, ta có:
\(\frac{a}{a+b}>\frac{a}{a+b+c};\frac{b}{b+c}>\frac{b}{a+b+c};\frac{c}{c+a}>\frac{c}{a+b+c}\)
\(\Rightarrow M>\frac{a}{a+b+c}+\frac{b}{a+b+c}+\frac{c}{a+b+c}=1\) (*)
Lại có: \(M=\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}=\frac{a+b-b}{a+b}+\frac{b+c-c}{b+c}+\frac{c+a-a}{c+a}=3-\left(\frac{b}{a+b}+\frac{c}{b+c}+\frac{a}{c+a}\right)\)
Chứng minh tương tự (*) ta có: \(\frac{b}{a+b}+\frac{c}{b+c}+\frac{a}{c+a}>1\)
\(\Rightarrow M< 3-1=2\) (**)
Từ (*) và (**) => 1 < M < 2 => đpcm
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có:\(\frac{b-c}{\left(a-b\right)\left(a-c\right)}=\frac{\left(a-c\right)-\left(a-b\right)}{\left(a-b\right)\left(a-c\right)}=\frac{a-c}{\left(a-b\right)\left(a-c\right)}-\frac{a-b}{\left(a-b\right)\left(a-c\right)}=\frac{1}{a-b}-\frac{1}{a-c}=\frac{1}{a-b}+\frac{1}{c-a}\left(1\right)\)Chứng minh tương tự,ta có:\(\hept{\begin{cases}\frac{c-a}{\left(b-c\right)\left(b-a\right)}=\frac{1}{b-c}+\frac{1}{a-b}\left(2\right)\\\frac{a-b}{\left(c-a\right)\left(c-b\right)}=\frac{1}{c-a}+\frac{1}{b-c}\left(3\right)\end{cases}}\)
Từ (1);(2);(3) suy ra:\(\frac{b-c}{\left(a-b\right)\left(a-c\right)}+\frac{c-a}{\left(b-c\right)\left(b-a\right)}+\frac{a-b}{\left(c-a\right)\left(c-b\right)}\)
\(=\frac{1}{a-b}+\frac{1}{c-a}+\frac{1}{b-c}+\frac{1}{a-b}+\frac{1}{c-a}+\frac{1}{b-c}\)
\(=2\left(\frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{c-a}\right)^{đpcm}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
2b = a+ c(1)
2bd = bc + bd
<=> ( a+c )d= bc+ cd
<=> ad +cd= bc+ cd
<=> ad = bc
<=> a/b = c/d (đpcm)
![](https://rs.olm.vn/images/avt/0.png?1311)
B1:
Từ \(b=\frac{a+c}{2}\Rightarrow2b=a+c\left(1\right)\)
Từ \(c=\frac{2bd}{b+a}\)thay vào (1) ta được:
\(2b=a+\frac{2bd}{b+a}\)
\(\Leftrightarrow2b\left(b+a\right)=a\left(b+a\right)+2bd\)
\(\Leftrightarrow2b^2+2ab=ab+a^2+2bd\)
\(\Leftrightarrow2b^2+ab-a^2-2bd=0\)
\(\Leftrightarrow2b\left(b-d\right)+a\left(b-a\right)=0\)
\(\Leftrightarrow2b\left(b-d\right)=a\left(a-b\right)\Leftrightarrow\frac{2b}{a}=\frac{a-b}{b-d}\)
B2: Từ \(\frac{1}{c}=\frac{1}{2}\left(\frac{1}{a}+\frac{1}{b}\right)\Rightarrow\frac{1}{c}=\frac{a+b}{2ab}hay2ab=c\left(a+b\right)\)
\(\Rightarrow ab+ab=ac+bc\Rightarrow ab-bc=ac-ab\Rightarrow b\left(a-c\right)=a\left(c-b\right)\)
Do đó: \(\frac{a-c}{c-b}=\frac{a}{b}\)(đpcm)
![](https://rs.olm.vn/images/avt/0.png?1311)
Gọi số dư của a và b khi chia m là n
Ta có: a=m*k+n
b=m*h+n
=>a-b=m*k+n -(m*h+n)
=m*k+n-m*h-n
=(m*k-m*h)+(n-n)
=m(k-h) luôn chia hết m
Đpcm
![](https://rs.olm.vn/images/avt/0.png?1311)
Đặt \(S=\frac{a}{a+b+c}+\frac{b}{b+c+d}+\frac{c}{c+d+a}+\frac{d}{d+a+b}\)
Ta có: \(\frac{a}{a+b+c}< \frac{a}{a+c}\)
\(\frac{b}{b+c+d}< \frac{b}{b+d}\)
\(\frac{c}{c+d+a}< \frac{c}{a+c}\)
\(\frac{d}{d+a+b}< \frac{d}{d+b}\)
\(\Rightarrow S< \left(\frac{a}{a+c}+\frac{c}{a+c}\right)+\left(\frac{b}{b+d}+\frac{d}{d+b}\right)\)
\(\Rightarrow S< 2\left(1\right)\)
Lại có: \(\frac{a}{a+b+c}>\frac{a}{a+b+c+d}\)
\(\frac{b}{b+c+d}>\frac{b}{b+c+a+d}\)
\(\frac{c}{c+d+a}>\frac{c}{a+b+c+d}\)
\(\frac{d}{d+a+b}>\frac{d}{a+b+c+d}\)
\(\Rightarrow S>1\left(2\right)\)
Từ (1) và (2) \(\Rightarrowđpcm\)
Theo đề ta có :
\(\frac{b}{a-c}=\frac{a+b}{c}=\frac{a}{b}\)
* Đầu tiên, ta xét
* \(\frac{b}{a-c}=\frac{a}{b}\):
\(\Rightarrow b^2=a\left(a-c\right)\) \(=a^2-ac\)
\(\Rightarrow a^2-b^2=ac\)(1)
* Xét \(\frac{a+b}{c}=\frac{a}{b}\)
\(\Rightarrow\left(a+b\right)b=ac\)
. Từ (1) ta thay \(ac=a^2-b^2\):
\(\Rightarrow\)\(\left(a+b\right)b=a^2-b^2\)
\(\Rightarrow\left(a+b\right)b=\left(a+b\right)\left(a-b\right)\)
\(\Rightarrow b=a-b\Rightarrow a=b+b=2b\)(2)
* Xét \(\frac{b}{a-c}=\frac{a+b}{c}\):
\(\Rightarrow bc=\left(a-c\right)\left(a+b\right)\)(với a = 2b)
\(\Rightarrow bc=\left(2b-c\right)\left(2b+b\right)\)
\(\Rightarrow bc=\left(2b-c\right).3b\)
\(\Rightarrow\frac{bc}{b}=\frac{\left(2b-c\right).3b}{b}\)
\(\Rightarrow c=\left(2b-c\right).3\)
\(\Rightarrow c=6b-3c\)
\(\Rightarrow6b=c+3c=4c\)(3)
Từ (2) và (3) \(\Rightarrow\)ta có :
\(a=2b\) và \(6b=4c\)
\(\Rightarrow\frac{a}{8}=\frac{b}{4}\)và \(\frac{b}{4}=\frac{c}{6}\)
\(\Rightarrow\frac{a}{8}=\frac{b}{4}=\frac{c}{6}\)(đpcm)
\(\frac{b}{a-c}=\frac{a+b}{c}=\frac{a}{b}=\frac{b+\left(a+b\right)+a}{a-c+c+b}=\frac{2\left(a+b+c\right)}{a+b+c}=2\)
\(\Rightarrow\frac{a}{b}=2\Leftrightarrow a=2b;\frac{a+b}{c}=2\Leftrightarrow a+b=2c\Leftrightarrow2b+b=2c\Leftrightarrow3b=2c\)
Ta có: \(\frac{a}{8}=\frac{2b}{8}=\frac{b}{4};\frac{c}{6}=\frac{2c}{12}=\frac{3b}{12}=\frac{b}{4}\)
=> \(\frac{a}{8}=\frac{b}{4}=\frac{c}{6}\)