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\(\text{Cho: }\frac{2y+2z-x}{a}=\frac{2z+2x-y}{b}=\frac{2z+2y-z}{c}\left(\text{tỉ lệ thức cuối sai sao lại có 2 lần 2z nếu là}\frac{2x+2y-z}{c}\right)\)
thì còn có thể hiểu đc!
2: Ta có: \(\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}=\dfrac{a\left(a+b+c\right)}{b+c}+\dfrac{b\left(a+b+c\right)}{c+a}+\dfrac{c\left(a+b+c\right)}{a+b}-a-b-c=\left(a+b+c\right)\left(\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\right)=a+b+c-a-b-c=0\)
1: Sửa đề: Cho \(x,y,z\ne0\) và \(\dfrac{1}{x}+\dfrac{2}{y}+\dfrac{1}{z}=\dfrac{2}{2x+y+2z}\).
CM:....
Đặt 2x = x', 2z = z'.
Ta có: \(\dfrac{2}{x'}+\dfrac{2}{y}+\dfrac{2}{z'}=\dfrac{2}{x'+y+z'}\)
\(\Leftrightarrow\dfrac{1}{x'}+\dfrac{1}{y}+\dfrac{1}{z'}=\dfrac{1}{x'+y+z'}\)
\(\Leftrightarrow\dfrac{1}{x'}-\dfrac{1}{x'+y+z'}+\dfrac{1}{y}+\dfrac{1}{z'}=0\)
\(\Leftrightarrow\dfrac{y+z'}{x'\left(x'+y+z'\right)}+\dfrac{y+z'}{yz'}=0\)
\(\Leftrightarrow\dfrac{\left(y+z'\right)\left(yz'+x'^2+x'y+x'z'\right)}{x'yz'\left(x'+y+z'\right)}=0\)
\(\Leftrightarrow\dfrac{\left(x'+y\right)\left(y+z'\right)\left(z'+x'\right)}{x'yz'\left(x'+y+z'\right)}=0\Leftrightarrow\left(2x+y\right)\left(y+2z\right)\left(2z+2x\right)=0\Leftrightarrow\left(2x+y\right)\left(y+2z\right)\left(z+x\right)=0\left(đpcm\right)\)
Ta đặt: \(\left\{{}\begin{matrix}\dfrac{1}{x^2}=a\\\dfrac{1}{y^2}=b\\\dfrac{1}{z^2}=c\end{matrix}\right.\)\(\Rightarrow\sqrt{abc}=abc=1\)
Ta có: \(\dfrac{1}{\sqrt{a}+\sqrt{ab}+1}+\dfrac{1}{\sqrt{b}+\sqrt{bc}+1}+\dfrac{1}{\sqrt{c}+\sqrt{ca}+1}\)
\(=\dfrac{1}{\sqrt{a}+\sqrt{ab}+1}+\dfrac{1}{\sqrt{b}+\dfrac{1}{\sqrt{a}}+1}+\dfrac{1}{\dfrac{1}{\sqrt{ab}}+\sqrt{ca}+1}\)
\(=\dfrac{1}{\sqrt{a}+\sqrt{ab}+1}+\dfrac{\sqrt{a}}{\sqrt{ba}+1+\sqrt{a}}+\dfrac{1}{1+\sqrt{ab}+\sqrt{a}}=1\)
Quay lại bài toán, sau khi đặt bài toán trở thành:
\(P=\dfrac{1}{2b+a+3}+\dfrac{1}{2c+b+3}+\dfrac{1}{2a+c+3}\)
\(=\dfrac{1}{\left(a+b\right)+\left(b+1\right)+2}+\dfrac{1}{\left(b+c\right)+\left(c+1\right)+2}+\dfrac{1}{\left(c+a\right)+\left(a+1\right)+2}\)
\(\le\dfrac{1}{2}\left(\dfrac{1}{\sqrt{a}+\sqrt{ab}+1}+\dfrac{1}{\sqrt{b}+\sqrt{bc}+1}+\dfrac{1}{\sqrt{c}+\sqrt{ca}+1}\right)=\dfrac{1}{2}\)
Cái đó t cố tình bỏ đấy. B phải tự làm chứ chẳng lẽ t làm hết??
Haha không giỡn nữa :v 
Áp dụng BĐT Cauchy-Schwarz ta có:
\(L.H.S=Σ\dfrac{1}{2x+y+z}=7Σ\dfrac{1}{2\left(x+3y\right)+\left(y+3z\right)+4\left(z+3x\right)}\)
\(=\dfrac{1}{7}Σ\dfrac{\left(2+1+4\right)^2}{2\left(x+3y\right)+\left(y+3z\right)+4\left(z+3x\right)}\)
\(\le\dfrac{1}{7}Σ\left(\dfrac{2^2}{2\left(x+3y\right)}+\dfrac{1^2}{y+3z}+\dfrac{4^2}{4\left(z+3x\right)}\right)\)
\(=\dfrac{1}{7}Σ\left(\dfrac{2}{x+3y}+\dfrac{1}{y+3z}+\dfrac{4}{z+3x}\right)\)
\(=\dfrac{1}{7}Σ\dfrac{7}{x+3y}=Σ\dfrac{1}{x+3y}=R.H.S\)
Áp dụng bất đẳng thức \(\dfrac{1}{x}+\dfrac{1}{y}\le\dfrac{4}{x+y}\) \(\forall x,y>0\)
\(\Rightarrow\left\{{}\begin{matrix}\dfrac{1}{x+3y}+\dfrac{1}{y+2z+x}\le\dfrac{4}{2x+4y+2z}=\dfrac{2}{x+2y+z}\\\dfrac{1}{y+3z}+\dfrac{1}{z+2x+y}\le\dfrac{4}{2x+2y+4z}=\dfrac{2}{x+y+2z}\\\dfrac{1}{z+3x}+\dfrac{1}{x+2y+z}\le\dfrac{4}{4x+2y+2z}=\dfrac{2}{2x+y+z}\end{matrix}\right.\)
\(\Rightarrow\dfrac{1}{x+3y}+\dfrac{1}{y+3z}+\dfrac{1}{z+3x}+\dfrac{1}{y+2z+x}+\dfrac{1}{z+2x+y}+\dfrac{1}{x+2y+z}\le\dfrac{2}{x+2y+z}+\dfrac{2}{x+y+2z}+\dfrac{2}{2x+y+z}\)
\(\Rightarrow VT\le\left(\dfrac{2}{x+2y+z}-\dfrac{1}{x+2y+z}\right)+\left(\dfrac{2}{x+y+2z}-\dfrac{1}{y+x+2z}\right)+\left(\dfrac{2}{2x+y+z}-\dfrac{1}{z+2x+y}\right)\)
\(\Rightarrow VT\le\dfrac{1}{x+2y+z}+\dfrac{1}{x+y+2z}+\dfrac{1}{2x+y+z}\)
\(\Leftrightarrow\dfrac{1}{x+3y}+\dfrac{1}{y+3z}+\dfrac{1}{z+3x}\le\dfrac{1}{x+2y+z}+\dfrac{1}{x+y+2z}+\dfrac{1}{2x+y+z}\) ( đpcm )
1.VT= \(\dfrac{x}{z}+\dfrac{y}{z}+\dfrac{y}{x}+\dfrac{z}{x}+\dfrac{z}{y}+\dfrac{x}{y}=\left(\dfrac{x}{y}+\dfrac{y}{x}\right)+\left(\dfrac{x}{z}+\dfrac{z}{x}\right)+\left(\dfrac{y}{z}+\dfrac{z}{y}\right)\)
Áp dụng BĐT Cô-si cho 2 số dương, ta có:
\(\dfrac{x}{y}+\dfrac{y}{x}\)≥ 2\(\sqrt{\dfrac{x}{y}.\dfrac{y}{x}}\)=2; tương tự \(\dfrac{x}{z}+\dfrac{z}{x}\)≥2; \(\dfrac{y}{z}+\dfrac{z}{y}\)≥2.
Cộng 3 BĐT trên, ta được đpcm.
2.Đặt b+c-a= x, a+c-b= y, a+b-c= z. Khi đó x,y,z>0.
2a= y+z; 2b= x+z; 2c= x+y. Khi đó bđt cần chứng minh trở thành:
\(\dfrac{x+y}{z}+\dfrac{y+z}{x}+\dfrac{z+x}{y}\)≥6.
Theo bài 1 bđt luôn đúng
Thay $x=\sqrt{\frac{1}{2,5}}; y=z=\sqrt{\frac{1}{0,25}}$ ta thấy đề sai bạn nhé!
\(VT=\dfrac{x^2}{x^2+2xy+3zx}+\dfrac{y^2}{y^2+2yz+3xy}+\dfrac{z^2}{z^2+2zx+3yz}\)
\(VT\ge\dfrac{\left(x+y+z\right)^2}{x^2+y^2+z^2+5xy+5yz+5zx}=\dfrac{\left(x+y+z\right)^2}{\left(x+y+z\right)^2+3\left(xy+yz+zx\right)}\ge\dfrac{\left(x+y+z\right)^2}{\left(x+y+z\right)^2+\left(x+y+z\right)^2}=\dfrac{1}{2}\)
\(\left(\dfrac{2x+2y-z}{3}\right)^2+\left(\dfrac{2y+2z-x}{3}\right)^2+\left(\dfrac{2z+2x-y}{3}\right)^2\)
\(=\dfrac{4y^2+4x^2+z^2+8xy-4xz-4yz+4y^2+4z^2+x^2+8yz-4xy-4xz}{9}+\dfrac{\left(2z+2x-y\right)^2}{9}\)
\(=\dfrac{8y^2+5x^2+5z^2+4xy-8xz+4yz+4z^2+4x^2+y^2+8xz-4yz-4xy}{9}\)
\(=\dfrac{9y^2+9z^2+9x^2}{9}=x^2+y^2+z^2\)
