a: \(\dfrac{2a+15b}{5a-7b}=\dfrac{2c+15d}{5c-7d}\)

\(\Leftrightarrow\left(2a+15b\right)\left(5c-7d\right)=\left(5a-7b\right)\left(2c+15d\right)\)

\(\Leftrightarrow10ac-14ad+75bc-105bd=10ac+75ad-14bc-105bd\)

\(\Leftrightarrow-14ad+75bc=-14bc+75ad\)

=>ad=bc

hay a/b=c/d

b: Đặt a/b=c/d=k

=>a=bk; c=dk

\(\dfrac{a^2}{b^2}=\dfrac{b^2k^2}{b^2}=k^2\)

\(\dfrac{2c^2-ac}{2d^2-bd}=\dfrac{2\cdot d^2k^2-bk\cdot dk}{2\cdot d^2-bd}=k^2\)

Do đó; \(\dfrac{a^2}{b^2}=\dfrac{2c^2-ac}{2d^2-bd}\)

1) Ta có:
\(\dfrac{a}{a+b}\)=\(\dfrac{c}{c+d}\)
=>a.(c+d) = c.(a+b)
a.c+a.d = a.c+b.d
Do đó a.d=b.d
=>\(\dfrac{a}{b}\)=\(\dfrac{c}{d}\)( đpcm)

Câu 2: 

Đặt a/b=c/d=k

=>a=bk; c=dk

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

\(\dfrac{-5a+3c}{-5b+3d}=\dfrac{-5bk+3dk}{-5b+3d}=k\)

=>\(\dfrac{3a+2c}{3b+2d}=\dfrac{-5a+3c}{-5b+3d}\)

b: \(\dfrac{a^2}{b^2}=\dfrac{b^2k^2}{b^2}=k^2\)

\(\dfrac{2c^2-ac}{2d^2-bd}=\dfrac{c\left(2c-a\right)}{d\left(2d-b\right)}=\dfrac{dk}{d}\cdot\dfrac{2dk-bk}{2d-b}=k^2\)

=>\(\dfrac{a^2}{b^2}=\dfrac{2c^2-ac}{2d^2-bd}\)

\(A=\frac{a}{ab+a+2}+\frac{b}{bc+b+1}+\frac{2c}{ac+2c+2}\)

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

\(=\frac{a}{a\left(bc+b+1\right)}+\frac{b}{bc+b+1}+\frac{abc^2}{ac\left(bc+b+1\right)}\)

\(=\frac{1}{bc+b+1}+\frac{b}{bc+b+1}+\frac{bc}{bc+b+1}\)

\(=\frac{bc+b+1}{bc+b+1}=1\)

Theo đề ra, ta có:

\(a^2+b^2+c^2\)

\(=\left(a+b+c\right)\left(a^2+b^2+c^2\right)\)

\(=a^3+b^3+c^3+a^2b+b^2c+c^2a+ab^2+bc^2+ca^2\)

Theo BĐT Cô-si:

\(\left\{{}\begin{matrix}a^3+ab^2\ge2a^2b\\b^3+bc^2\ge2b^2c\\c^3+ca^2\ge2c^2a\end{matrix}\right.\Rightarrow a^2+b^2+c^2\ge3\left(a^2b+b^2c+c^2a\right)\)

Do vậy \(M\ge14\left(a^2+b^2+c^2\right)+\dfrac{3\left(ab+bc+ac\right)}{a^2+b^2+c^2}\)

Ta đặt \(a^2+b^2+c^2=k\)

Luôn có \(3\left(a^2+b^2+c^2\right)\ge\left(a+b+c\right)^2=1\)

Vì thế nên \(k\ge\dfrac{1}{3}\)

Khi đấy:

\(M\ge14k+\dfrac{3\left(1-k\right)}{2k}=\dfrac{k}{2}+\dfrac{27k}{2}+\dfrac{3}{2k}-\dfrac{3}{2}\ge\dfrac{1}{3}.\dfrac{1}{2}+2\sqrt{\dfrac{27k}{2}.\dfrac{3}{2k}}-\dfrac{3}{2}=\dfrac{23}{3}\)

\(\Rightarrow Min_M=\dfrac{23}{3}\Leftrightarrow a=b=c=\dfrac{1}{3}\).

Hint: \(M=3-Σ\dfrac{a^2+a+1}{a^2+2a+1}\)

dự đoán dấu đẳng thức xảy ra tại \(a=b=c=\sqrt{3}-1\) nên \(MinM=\dfrac{3\sqrt{3}}{4}\)ta đi chứng minh \(M\le\dfrac{3\sqrt{3}}{4}\)

Đặt \(a+1=x\); \(b+1=y\); \(c+1=z\)

\(\Rightarrow x+y+z=a+b+c+3=a+b+c+ab+bc+ac+abc+1\)

\(=\left(a+1\right)\left(b+1\right)\left(c+1\right)\)\(=xyz\)

\(\Rightarrow x+y+z=xyz\) đặt \(x=tan\dfrac{A}{2};y=tan\dfrac{B}{2};z=tan\dfrac{C}{2}\)

\(\Rightarrow tan\dfrac{A}{2}+tan\dfrac{B}{2}+tan\dfrac{C}{2}=tan\dfrac{A}{2}.tan\dfrac{B}{2}.tan\dfrac{C}{2}\)(đúng , tự cm hoặc google)

do \(x=tan\dfrac{A}{2}\Rightarrow sinA=\dfrac{2x}{x^2+1}\Rightarrow\dfrac{sinA}{2}=\dfrac{x}{x^2+1}\)

\(\Rightarrow M=\Sigma\dfrac{sinA}{2}\)ta phai cm \(M=\Sigma\dfrac{sinA}{2}\le\dfrac{3\sqrt{3}}{4}\Leftrightarrow M\le\dfrac{3\sqrt{3}}{2}\)

\(\Leftrightarrow sinA+sinB+sinC\le\dfrac{3\sqrt{3}}{2}\)theo BDT Cauchy-Swarch

\(\Rightarrow sinA+sinB+sinC\le\sqrt{3\left(sinA^2+sin^2B+sin^2C\right)}\)

mặt khác \(sin^2A+sin^2B+sin^2C\le\dfrac{9}{4}\)(trong sach toan 10 co BDT nay hoac google)

\(\Rightarrow sinA+sinB+sinC\le\sqrt{\dfrac{3.9}{4}}=\dfrac{3\sqrt{3}}{2}\)

\(\Rightarrow dpcm\)