$(x;y;z)=\{(6;9;12);(8;12;16)\}$

Giải thích các bước giải:

$\begin{array}{c}\frac{2z-4x}{3}=\frac{3x-2y}{4}=\frac{4y-3z}{2}\\ \Rightarrow \frac{3(2z-4x)}{9}=\frac{4(3x-2y)}{16}=\frac{2(4y-3z)}{4}\\ =\frac{6z-12x+12x-8y+8y-6z}{9+16+4}=0\end{array}$

$\Rightarrow \{\begin{array}{c}2z-4x=0\\ 3x-2y=0\\ 4y-3z=0\end{array}\Rightarrow y=\frac{3}{4}z$

mà $200<y^2+z^2<450$

$\begin{array}{c}\Rightarrow 200<{\left(\frac{3}{4}z\right)}^2+z^2<450\\ \iff 200<\frac{25}{16}{z}^2<450\\ \iff 128<z^2<288\end{array}$

Vì z là số nguyên dương $\Rightarrow \sqrt{128}<z<\sqrt{288}$

$\Rightarrow z\in \{12;13;14;15;16\}$

mà y là số nguyên dương và $y=\frac{3}{4}z$

$\Rightarrow z\in \{12;16\}$

Thế vào $y=\frac{3}{4}z$ và $2z-4x=0$

+) Với $z=12\Rightarrow y=\frac{3}{4}.12=6$

                    $2.12-4x=0\Rightarrow x=6$

Với $z=16\Rightarrow y=\frac{3}{4}.16=12$

    $2.16-4x=0\Rightarrow x=8$

Vậy ta có các cặp nghiệm là: $(x;y;z)=\{(6;9;12);(8;12;16)\}$

$(x;y;z)=\{(6;9;12);(8;12;16)\}$

Giải thích các bước giải:

$\begin{array}{c}\frac{2z-4x}{3}=\frac{3x-2y}{4}=\frac{4y-3z}{2}\\ \Rightarrow \frac{3(2z-4x)}{9}=\frac{4(3x-2y)}{16}=\frac{2(4y-3z)}{4}\\ =\frac{6z-12x+12x-8y+8y-6z}{9+16+4}=0\end{array}$

$\Rightarrow \{\begin{array}{c}2z-4x=0\\ 3x-2y=0\\ 4y-3z=0\end{array}\Rightarrow y=\frac{3}{4}z$

mà $200<y^2+z^2<450$

$\begin{array}{c}\Rightarrow 200<{\left(\frac{3}{4}z\right)}^2+z^2<450\\ \iff 200<\frac{25}{16}{z}^2<450\\ \iff 128<z^2<288\end{array}$

Vì z là số nguyên dương $\Rightarrow \sqrt{128}<z<\sqrt{288}$

$\Rightarrow z\in \{12;13;14;15;16\}$

mà y là số nguyên dương và $y=\frac{3}{4}z$

$\Rightarrow z\in \{12;16\}$

Thế vào $y=\frac{3}{4}z$ và $2z-4x=0$

+) Với $z=12\Rightarrow y=\frac{3}{4}.12=6$

                    $2.12-4x=0\Rightarrow x=6$

Với $z=16\Rightarrow y=\frac{3}{4}.16=12$

    $2.16-4x=0\Rightarrow x=8$

Vậy ta có các cặp nghiệm là: $(x;y;z)=\{(6;9;12);(8;12;16)\}$

a) \(\left\{{}\begin{matrix}a=x\\b=2y\\c=3z\end{matrix}\right.\Rightarrow a+b+c=2;a,b,c>0\)

\(\Rightarrow S=\sqrt{\dfrac{\dfrac{ab}{2}}{\dfrac{ab}{2}+c}}+\sqrt{\dfrac{\dfrac{bc}{2}}{\dfrac{bc}{2}+a}}+\sqrt{\dfrac{ca}{ca+2b}}\)

\(=\sqrt{\dfrac{ab}{ab+2c}}+\sqrt{\dfrac{bc}{bc+2a}}+\sqrt{\dfrac{ca}{ca+2b}}\)

Vì a,b,c>0 nên áp dụng BĐT AM-GM, ta có: 

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

\(=\sqrt{\dfrac{a}{a+c}}.\sqrt{\dfrac{b}{b+c}}\le\dfrac{1}{2}\left(\dfrac{a}{a+c}+\dfrac{b}{b+c}\right)\) 

\(\sqrt{\dfrac{bc}{bc+2a}}=\sqrt{\dfrac{bc}{\left(b+a\right)\left(c+a\right)}}\le\dfrac{1}{2}\left(\dfrac{b}{a+b}+\dfrac{c}{a+c}\right)\)

\(\sqrt{\dfrac{ca}{ca+2b}}=\sqrt{\dfrac{ca}{\left(c+b\right)\left(a+b\right)}}\le\dfrac{1}{2}\left(\dfrac{c}{b+c}+\dfrac{a}{a+b}\right)\)

\(\Rightarrow S\le\dfrac{1}{2}\left(\dfrac{a}{a+b}+\dfrac{b}{a+b}\right)+\dfrac{1}{2}\left(\dfrac{b}{b+c}+\dfrac{c}{b+c}\right)+\dfrac{1}{2}\left(\dfrac{a}{a+c}+\dfrac{c}{a+c}\right)=\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}=\dfrac{3}{2}\)

Dấu "=" xảy ra khi và chỉ khi: a=b=c=2/3=>\(\left(x,y,z\right)=\left\{\dfrac{2}{3};\dfrac{1}{3};\dfrac{2}{9}\right\}\)

Đặt \(\left(x;2y;3z\right)=\left(a;b;c\right)\Rightarrow a+b+c=2\)

\(S=\sqrt{\dfrac{ab}{ab+2c}}+\sqrt{\dfrac{bc}{bc+2a}}+\sqrt{\dfrac{ca}{ca+2b}}\)

\(S=\sqrt{\dfrac{ab}{ab+c\left(a+b+c\right)}}+\sqrt{\dfrac{bc}{bc+a\left(a+b+c\right)}}+\sqrt{\dfrac{ca}{ca+b\left(a+b+c\right)}}\)

\(S=\sqrt{\dfrac{ab}{\left(a+c\right)\left(b+c\right)}}+\sqrt{\dfrac{bc}{\left(a+b\right)\left(a+c\right)}}+\sqrt{\dfrac{ca}{\left(a+b\right)\left(b+c\right)}}\)

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

Dấu "=" xảy ra khi \(a=b=c=\dfrac{2}{3}\Rightarrow x;y;z\)