3/ Áp dụng bất đẳng thức AM-GM, ta có :

\(\dfrac{a^2}{b^2}+\dfrac{b^2}{c^2}\ge2\sqrt{\dfrac{\left(ab\right)^2}{\left(bc\right)^2}}=\dfrac{2a}{c}\)

\(\dfrac{b^2}{c^2}+\dfrac{c^2}{a^2}\ge2\sqrt{\dfrac{\left(bc\right)^2}{\left(ac\right)^2}}=\dfrac{2b}{a}\)

\(\dfrac{c^2}{a^2}+\dfrac{a^2}{b^2}\ge2\sqrt{\dfrac{\left(ac\right)^2}{\left(ab\right)^2}}=\dfrac{2c}{b}\)

Cộng 3 vế của BĐT trên ta có :

\(2\left(\dfrac{a^2}{b^2}+\dfrac{b^2}{c^2}+\dfrac{c^2}{a^2}\right)\ge2\left(\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}\right)\)

\(\Leftrightarrow\dfrac{a^2}{b^2}+\dfrac{b^2}{c^2}+\dfrac{c^2}{a^2}\ge\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}\left(\text{đpcm}\right)\)

Bài 1:

Áp dụng BĐT AM-GM ta có:

\(\frac{1}{a^2+bc}+\frac{1}{b^2+ac}+\frac{1}{c^2+ab}\leq \frac{1}{2\sqrt{a^2.bc}}+\frac{1}{2\sqrt{b^2.ac}}+\frac{1}{2\sqrt{c^2.ab}}=\frac{\sqrt{ab}+\sqrt{bc}+\sqrt{ac}}{2abc}\)

Tiếp tục áp dụng BĐT AM-GM:

\(\sqrt{bc}+\sqrt{ac}+\sqrt{ab}\leq \frac{b+c}{2}+\frac{c+a}{2}+\frac{a+b}{2}=a+b+c\)

Do đó:

\(\frac{1}{a^2+bc}+\frac{1}{b^2+ac}+\frac{1}{c^2+ab}\leq \frac{\sqrt{ab}+\sqrt{bc}+\sqrt{ca}}{2abc}\leq \frac{a+b+c}{2abc}\) (đpcm)

Dấu "=" xảy ra khi $a=b=c$

Bài 2:

Thay $1=a+b+c$ và áp dụng BĐT AM-GM ta có:

\(\left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right)\left(1+\frac{1}{c}\right)=\frac{(a+1)(b+1)(c+1)}{abc}\)

\(=\frac{(a+a+b+c)(b+a+b+c)(c+a+b+c)}{abc}\)

\(\geq \frac{4\sqrt[4]{a.a.b.c}.4\sqrt[4]{b.a.b.c}.4\sqrt[4]{c.a.b.c}}{abc}=\frac{64abc}{abc}=64\)

Ta có đpcm
Dấu "=" xảy ra khi $a=b=c=\frac{1}{3}$

\(VT=\dfrac{a}{c+b}+\dfrac{b}{a+c}+\dfrac{c}{a+b}=\dfrac{a}{c+b}+1+\dfrac{b}{a+c}+1+\dfrac{c}{a+b}-3=\dfrac{a+b+c}{b+c}+\dfrac{a+b+c}{a+c}+\dfrac{a+b+c}{a+b}-3=\left(a+b+c\right)\left(\dfrac{1}{b+c}+\dfrac{1}{a+c}+\dfrac{1}{a+b}\right)-3\)

-Áp dụng BĐT Caushy Schwarz cho 3 số dương ta có:

\(VT\ge\left(a+b+c\right).\dfrac{\left(1+1+1\right)^2}{a+b+b+c+c+a}-3=\left(a+b+c\right).\dfrac{9}{2\left(a+b+c\right)}-3=\dfrac{9}{2}-3=\dfrac{3}{2}\left(1\right)\)

\(VP=\dfrac{2.\left(\dfrac{a}{a^2+1}+\dfrac{1}{2}+\dfrac{b}{b^2+1}+\dfrac{1}{2}+\dfrac{c}{c^2+1}+\dfrac{1}{2}-\dfrac{3}{2}\right)}{2}=\dfrac{\dfrac{2a}{a^2+1}+1+\dfrac{2b}{b^2+1}+1+\dfrac{c}{c^2+1}-3}{2}=\dfrac{\dfrac{a^2+2a+1}{a^2+1}+\dfrac{b^2+2b+1}{b^2+1}+\dfrac{c^2+2c+1}{c^2+1}-3}{2}=\dfrac{\dfrac{\left(a+1\right)^2}{a^2+1}+\dfrac{\left(b+1\right)^2}{b^2+1}+\dfrac{\left(c+1\right)^2}{c^2+1}-3}{2}\)-Áp dụng BĐT Caushy ta có:

\(VP\le\dfrac{\dfrac{2\left(a^2+1\right)}{a^2+1}+\dfrac{2\left(b^2+1\right)}{b^2+1}+\dfrac{2\left(c^2+1\right)}{c^2+1}-3}{2}=\dfrac{2+2+2-3}{2}=\dfrac{3}{2}\left(2\right)\)

-Từ (1) và (2) ta có:

\(VT\ge\dfrac{3}{2}\ge VP\Rightarrow\dfrac{a}{c+b}+\dfrac{b}{a+c}+\dfrac{c}{a+b}\ge\dfrac{3}{2}\ge\dfrac{a}{a^2+1}+\dfrac{b}{b^2+1}+\dfrac{c}{c^2+1}\left(đpcm\right)\)

-Dấu bằng xảy ra \(\Leftrightarrow a=b=c=1\)

a. \(a^3+a^2c-abc+b^2c+b^3\)

<=> \(\left(a^3+b^3\right)+c\left(a^2-ab+b^2\right)\)

<=> (\(\left(a+b\right)\left(a^2-ab+b^2\right)+c\left(a^2-ab+b^2\right)\)

<=> \(\left(a+b+c\right)\left(a^2-ab+b^2\right)\)

vì a+b+c =0 => đpcm

b. 2(a+1)(b+1)=(a+b)(a+b+2)

<=> \(2\left(ab+a+b+1\right)=\)\(a^2+ab+2a+ab+b^2+2b\)

<=> \(2ab+2a+2b+2=a^2ab+2a+ab+b^2+2b\)

<=> \(a^2+b^2=2\)=> đpcm

a. a^3+a^2c-abc+b^2c+b^3a3+a2c−abc+b2c+b3

<=> \left(a^3+b^3\right)+c\left(a^2-ab+b^2\right)(a3+b3)+c(a2−ab+b2)

<=> (\left(a+b\right)\left(a^2-ab+b^2\right)+c\left(a^2-ab+b^2\right)(a+b)(a2−ab+b2)+c(a2−ab+b2)

<=> \left(a+b+c\right)\left(a^2-ab+b^2\right)(a+b+c)(a2−ab+b2)

vì a+b+c =0 => đpcm

b. 2(a+1)(b+1)=(a+b)(a+b+2)

<=> 2\left(ab+a+b+1\right)=2(ab+a+b+1)=a^2+ab+2a+ab+b^2+2ba2+ab+2a+ab+b2+2b

<=> 2ab+2a+2b+2=a^2ab+2a+ab+b^2+2b2ab+2a+2b+2=a2ab+2a+ab+b2+2b

<=> a^2+b^2=2a2+b2=2=> đpcm

Bạn tham khảo thêm tại đây:

Câu hỏi của Cậu Bé Ngu Ngơ - Toán lớp 8 | Học trực tuyến

Lời giải:
$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=1$

$\Leftrightarrow (a+b+c)\left(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\right)=a+b+c$

$\Leftrightarrow \frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}+\frac{ab+bc}{c+a}+\frac{ac+bc}{a+b}+\frac{ab+ac}{b+c}=a+b+c$

$\Leftrightarrow \frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}+b+c+a=a+b+c$

$\Leftrightarrow \frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}=0$

Ta có đpcm.

Câu 1 

Ta có : \(\frac{a}{b}=\frac{c}{d}=>\left(\frac{a}{b}+1\right)=\left(\frac{c}{d}+1\right)\left(=\right)\frac{a+b}{b}=\frac{c+d}{d}\)

=> ĐPCM

Câu 2

Ta có \(\frac{a}{b}=\frac{c}{d}=>\frac{b}{a}=\frac{d}{c}=>\left(\frac{b}{a}+1\right)=\left(\frac{d}{c}+1\right)\left(=\right)\frac{b+a}{a}=\frac{d+c}{c}=>\frac{a}{b+a}=\frac{c}{d+c}\)

=> ĐPCM

Câu 3

Câu 3

Ta có \(\frac{a+b}{a-b}=\frac{c+d}{c-d}\)(=) (a+b).(c-d)=(a-b).(c+d)(=)ac-ad+bc-bd=ac+ad-bc-bd(=)-ad+bc=ad-bc(=) bc+bc=ad+ad(=)2bc=2ad(=)bc=ad=> \(\frac{a}{b}=\frac{c}{d}\)

=> ĐPCM

Câu 4 

Đặt \(\frac{a}{b}=\frac{c}{d}=k\)

\(=>\hept{\begin{cases}a=bk\\c=dk\end{cases}}\)

Ta có \(\frac{ac}{bd}=\frac{bk.dk}{bd}=k^2\left(1\right)\)

Lại có \(\frac{a^2+c^2}{b^2+d^2}=\frac{b^2k^2+c^2k^2}{b^2+d^2}=\frac{k^2.\left(b^2+d^2\right)}{b^2+d^2}=k^2\left(2\right)\)

Từ (1) và (2) => ĐPCM

Mày là thằng anh tuấn lớp 7c trường THCS yên lập đúng ko