Đặt \(\dfrac{a}{b}=\dfrac{c}{d}=k\)

=>\(a=bk;c=dk\)

1: \(\dfrac{2a+3c}{2b+3d}=\dfrac{2\cdot bk+3\cdot dk}{2b+3d}=\dfrac{k\left(2b+3d\right)}{2b+3d}=k\)

\(\dfrac{2a-3c}{2b-3d}=\dfrac{2bk-3dk}{2b-3d}=\dfrac{k\left(2b-3d\right)}{2b-3d}=k\)

Do đó: \(\dfrac{2a+3c}{2b+3d}=\dfrac{2a-3c}{2b-3d}\)

2: \(\dfrac{4a-3b}{4c-3d}=\dfrac{4\cdot bk-3b}{4\cdot dk-3d}=\dfrac{b\left(4k-3\right)}{d\left(4k-3\right)}=\dfrac{b}{d}\)

\(\dfrac{4a+3b}{4c+3d}=\dfrac{4bk+3b}{4dk+3d}=\dfrac{b\left(4k+3\right)}{d\left(4k+3\right)}=\dfrac{b}{d}\)

Do đó: \(\dfrac{4a-3b}{4c-3d}=\dfrac{4a+3b}{4c+3d}\)

3: \(\dfrac{3a+5b}{3a-5b}=\dfrac{3bk+5b}{3bk-5b}=\dfrac{b\left(3k+5\right)}{b\left(3k-5\right)}=\dfrac{3k+5}{3k-5}\)

\(\dfrac{3c+5d}{3c-5d}=\dfrac{3dk+5d}{3dk-5d}=\dfrac{d\left(3k+5\right)}{d\left(3k-5\right)}=\dfrac{3k+5}{3k-5}\)

Do đó: \(\dfrac{3a+5b}{3a-5b}=\dfrac{3c+5d}{3c-5d}\)

4: \(\dfrac{3a-7b}{b}=\dfrac{3bk-7b}{b}=\dfrac{b\left(3k-7\right)}{b}=3k-7\)

\(\dfrac{3c-7d}{d}=\dfrac{3dk-7d}{d}=\dfrac{d\left(3k-7\right)}{d}=3k-7\)

Do đó: \(\dfrac{3a-7b}{b}=\dfrac{3c-7d}{d}\)

Ta có :

\(\dfrac{2a+3c}{2b+3d}=\dfrac{2a-3c}{2b-3d}\)

\(\Leftrightarrow\dfrac{2a}{2b}=\dfrac{3c}{3d}=\dfrac{2a}{2b}=\dfrac{3c}{3d}\) (Áp dụng t/c dãy tỉ số bằng nhau)

\(\Leftrightarrow\dfrac{a}{b}=\dfrac{c}{d}=\dfrac{a}{b}=\dfrac{c}{d}\)

\(\Leftrightarrow\dfrac{a}{b}=\dfrac{c}{d}\left(đpcm\right)\)

Đặt \(\dfrac{a}{b}=\dfrac{c}{d}=k\)

\(\Leftrightarrow\left\{{}\begin{matrix}a=bk\\c=dk\end{matrix}\right.\)

\(\dfrac{2a+3c}{3a+4c}=\dfrac{2bk+3dk}{3bk+4dk}=\dfrac{2b+3d}{3b+4d}\)

\(\dfrac{a}{b}=\dfrac{c}{d}\Leftrightarrow ad=bc\)

Ta có:

Nếu:

\(\dfrac{2a+c}{2b+d}=\dfrac{a-c}{b-d}\Leftrightarrow\left(2a+c\right)\left(b-d\right)=\left(a-c\right)\left(2b+d\right)\)

\(\Leftrightarrow2a\left(b-d\right)+c\left(b-d\right)=a\left(2b+d\right)-c\left(2b+d\right)\)

\(\Leftrightarrow2ab-2ad+bc-cd=2ab+ad-2bc+cd\)

\(\Leftrightarrow ad=bc\)

\(\Leftrightarrow\dfrac{2a+c}{2b+d}=\dfrac{a-c}{b-d}\left(đpcm\right)\)

Đặt \(\dfrac{a}{b}=\dfrac{c}{d}=k\)

\(\Rightarrow\left\{{}\begin{matrix}a=bk\\c=dk\end{matrix}\right.\) (1)

Thay (1) vào đề:

\(VT=\left(2a+3c\right)\left(b+d\right)=\left(2bk+3dk\right)\left(b+d\right)=2b^2k+3bdk+2bdk+3d^2k=3d^2k+2b^2k+5bdk\)

\(VP=\left(bk+dk\right)\left(2b+3d\right)=2b^2k+2bdk+3bdk+3d^2k=3d^2k+2b^2k+5bdk\)

Khi đó: \(VT=VP\)

\(\Leftrightarrow\left(2a+3c\right)\left(b+d\right)=\left(a+c\right)\left(2b+3d\right)\rightarrowđpcm.\)

Đặt \(\dfrac{a}{b}=\dfrac{c}{d}=k\Rightarrow\left\{{}\begin{matrix}a=bk\\c=dk\end{matrix}\right.\)

Ta có:

\(\left(2a+3c\right)\left(b+d\right)=\left(2bk+3dk\right)\left(b+d\right)=2b^2k+2bkd+3bkd+3d^2k\)

\(=2b^2k+5bkd+3d^2k\)(1)

\(\left(a+c\right)\left(2b+3d\right)=\left(bk+dk\right)\left(2b+3d\right)=2b^2k+3bkd+2bkd+3d^2k\)

\(=2b^2k+5bkd+3d^2k\)(2)

Từ (1) và (2) suy ra:

\(\left(2a+3c\right).\left(b+d\right)=\left(a+c\right)\left(2b+3d\right)\)(đpcm)

Chúc bạn học tốt!!!

Các bạn chỉ cần giúp mk câu b, c, e, f,

bạn cứ đặt công thức gốc là k sau đó thay vào các câu là được thui

Áp dụng BĐT Cauchy-Schwarz dạng Engel ta có:

\(VT=\dfrac{a}{b+2c+3d}+\dfrac{b}{c+2d+3a}+\dfrac{c}{d+2a+3b}+\dfrac{d}{a+2b+3c}\)

\(=\dfrac{a^2}{ab+2ac+3ad}+\dfrac{b^2}{bc+2bd+3ab}+\dfrac{c^2}{cd+2ac+3bc}+\dfrac{d^2}{ad+2bd+3cd}\)

\(\ge\dfrac{\left(a+b+c+d\right)^2}{4\left(ab+ad+bc+bd+ca+cd\right)}\ge\dfrac{\left(a+b+c+d\right)^2}{\dfrac{3}{2}\left(a+b+c+d\right)^2}=\dfrac{2}{3}\)

*Chứng minh \(4\left(ab+ad+bc+bd+ca+cd\right)\le\dfrac{3}{2}\left(a+b+c+d\right)^2\)

\(\Leftrightarrow\left(a-b\right)^2+\left(a-d\right)^2+\left(b-c\right)^2+\left(b-d\right)^2+\left(a-c\right)^2+\left(c-d\right)^2\ge0\)

Làm lại lun ._.