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![](https://rs.olm.vn/images/avt/0.png?1311)
Có: \(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}=\frac{1}{2019}\)
\(\Leftrightarrow\left(a+b+c\right)\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)=2019.\frac{1}{2019}\)
\(\Leftrightarrow1+\frac{c}{a+b}+1+\frac{a}{b+c}+1+\frac{b}{a+c}=1\)
\(\Leftrightarrow\frac{c}{a+b}+\frac{a}{b+c}+\frac{b}{a+c}=-2\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\frac{2\left|2018x-2019\right|+2019}{\left|2018x-2019\right|+1}\)
\(=\frac{\left(2\left(\left|2018x-2019\right|+1\right)\right)+2017}{\left|2018x-2019\right|+1}\)
\(=2+\frac{2017}{\left|2018x-2019\right|+1}\)có giá trị lớn nhất
\(\Rightarrow\frac{2017}{\left|2018x-2019\right|+1}\)có giá trị lớn nhất
\(\Rightarrow\left|2018x-2019\right|+1\)có giá trị nhỏ nhất
Mà \(\left|2018x-2019\right|\ge0\)
\(\Rightarrow\left|2018x-2019\right|+1\ge1\)
Dấu "=" xảy ra khi và chỉ khi:
\(\left|2018x-2019\right|=0\)
\(\Leftrightarrow x=\frac{2019}{2018}\)
Vậy \(M_{MAX}=2019\)tại \(x=\frac{2019}{2018}\)
\(\frac{5^x+5^{x+1}+5^{x+2}}{31}=\frac{3^{2x}+3^{2x+1}+3^{2x+2}}{13}\)
\(\Rightarrow\frac{5^x\left(1+5+5^2\right)}{31}=\frac{3^{2x}\left(1+3+3^2\right)}{13}\)
\(\Rightarrow\frac{5^x\cdot31}{31}=\frac{3^{2x}\cdot13}{13}\)
\(\Rightarrow5^x=3^{2x}\)
Mà \(\left(5;3\right)=1\)
\(\Rightarrow x=2x=0\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có : \(P=\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\)
\(\Rightarrow P+3=\frac{a}{b+c}+1+\frac{b}{c+a}+1+\frac{c}{a+b}+1\)
\(\Rightarrow P+3=\frac{a+b+c}{b+c}+\frac{a+b+c}{c+a}+\frac{a+b+c}{a+b}\)
\(\Rightarrow P+3=\left(a+b+c\right).\frac{1}{b+c}+\left(a+b+c\right).\frac{1}{c+a}+\left(a+b+c\right).\frac{1}{a+b}\)
\(\Rightarrow P+3=\left(a+b+c\right).\left(\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{a+b}\right)\)
\(\Rightarrow P+3=2019.10\)
\(\Rightarrow P+3=20190\)
\(\Rightarrow P=20190-3\)
\(\Rightarrow P=20187\)
Vậy P = 20187
![](https://rs.olm.vn/images/avt/0.png?1311)
Theo đề: \(\left(a+b+c\right)\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)=\frac{2019}{90}\)
Khai triển:
\(\left(a+b+c\right)\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)\)
\(=\frac{a}{a+b}+\frac{a}{b+c}+\frac{a}{c+a}+\frac{b}{a+b}+\frac{b}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\frac{c}{b+c}+\frac{c}{a+c}\)
\(=\frac{a+b}{a+b}+\frac{a+c}{a+c}+\frac{b+c}{b+c}+\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\)
\(\Leftrightarrow\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+3=\frac{2019}{90}\)
Làm nốt nhé :3
![](https://rs.olm.vn/images/avt/0.png?1311)
\(a+b+c = 1 ; 1/a + 1/b + 1/c = 1 \)
\(=> (a+b+c)(1/a +1/b+1/c) = 1\)
\(<=> a/b + b/a + a/c + c/a + b/c + c/b + 3 - 1 = 0\)
\(<=> (a^2+b^2)/ab + (a^2+c^2)/ac + (b^2+c^2)/bc + 2 =0\)
\(<=> (a^2 + b^2).c + (a^2+c^2).b + (b^2+c^2).a + 2abc = 0\)
\(<=> a^2c + b^2c + a^2b + c^2b + ab^2 + ac^2 + 2abc =0 \)
\(<=> a^2c + ac^2 + abc + a^2b+ ab^2 + abc + b^2c + bc^2 =0\)
\(<=> ac(a+b+c) + ab(a+b+c) + bc(b+c) =0 \)
\(<=> a(b+c)(a+b+c) + bc(b+c) =0 \)
\(<=> (b+c)(a^2 + ab + ac + bc ) = 0 \)
\(<=> (b+c)[a(a+b) + c(a+b)] =0\)
\(<=> (b+c)(a+b)(a+c) =0 \)
<=> 1 trong 3 số \(b+c;a+b ; a+c = 0\)
\(a+b=0 => a= -b => a + b + c = 1 <=> c = 1 ; a = b = 0\)
Thay vào S ta được : \(\Rightarrow S=0^{2019}+0^{2019}+1^{2019}=1\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Bài giải
* Từ \(\frac{a}{b}=\frac{c}{d}\text{ }\Rightarrow\text{ }\frac{a}{c}=\frac{b}{d}\text{ }\Rightarrow\text{ }\frac{a^{2019}}{c^{2019}}=\frac{b^{2019}}{d^{2019}}=\frac{a^{2019}+b^{2019}}{c^{2019}+d^{2019}}\text{ ( * ) }\)
* Từ \(\frac{a}{c}=\frac{b}{d}=\frac{a-b}{c-d}\text{ }\Rightarrow\text{ }\frac{a^{2019}}{c^{2019}}=\frac{\left(a-b\right)^{2019}}{\left(c-d\right)^{2019}}\left(\text{**}\right)\)
* Từ \(\left(\text{*}\right),\left(\text{**}\right)\Rightarrow\text{ ĐPCM}\)
ta có \(\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{b+c}\)
=\(\frac{a}{b+c}+1+\frac{b}{a+c}+1+\frac{c}{b+c}+1-3\)
=\(\frac{a+b+c}{b+c}+\frac{a+b+c}{a+c}+\frac{a+b+c}{a+b}-3\)
=\(\left(a+b+c\right)\left(\frac{1}{c+b}+\frac{1}{a+c}+\frac{1}{a+b}\right)-3\)
rồi còn lại thay vào nha bn
\(\left(a+b+c\right)\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)=2019\cdot\frac{1}{2019}\)
\(\Leftrightarrow\frac{a+b+c}{a+b}+\frac{a+b+c}{b+c}+\frac{a+b+c}{c+a}=1\)
\(\Leftrightarrow\left(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\right)+3=1\)
\(\Leftrightarrow S=-2\)