Đặt \(H=\frac{xz}{y^2+yz}+\frac{y^2}{zx+yz}+\frac{x+2z}{x+z}\)

\(=\frac{1}{\frac{y^2}{xz}+\frac{yz}{xz}}+\frac{1}{\frac{zx}{y^2}+\frac{yz}{y^2}}+\frac{x+z+z}{x+z}\)

\(=\frac{1}{\frac{y^2}{zx}+\frac{y}{x}}+\frac{1}{\frac{zx}{y^2}+\frac{z}{y}}+\frac{1}{\frac{x}{z}+1}+1\)

Đặt \(\frac{x}{y}=a;\frac{y}{z}=b\Rightarrow ab=\frac{x}{z}\ge1\)

Khi đó \(H=\frac{1}{\frac{b}{a}+\frac{1}{a}}+\frac{1}{\frac{a}{b}+\frac{1}{b}}+\frac{1}{ab+1}+1\)

\(=\frac{a}{b+1}+\frac{b}{a+b}+\frac{1}{ab+1}+1\)

Ta cần chứng minh \(U=\frac{a}{b+c}+\frac{b}{a+b}+\frac{1}{ab+1}\ge\frac{3}{2}\)

\(\Leftrightarrow\left(\frac{a}{b+1}+1\right)+\left(\frac{b}{a+1}+1\right)+\frac{1}{ab+1}\ge\frac{7}{2}\)

\(\Leftrightarrow\frac{a+b+1}{b+1}+\frac{a+b+1}{a+1}+\frac{1}{ab+1}\ge\frac{7}{2}\)

\(\Leftrightarrow\left(a+b+1\right)\left(\frac{1}{b+1}+\frac{1}{a+1}\right)+\frac{1}{ab+1}\ge\frac{7}{2}\)

Khi đó \(Y=\left(a+b+1\right)\left(\frac{1}{a+1}+\frac{1}{b+1}\right)+\frac{1}{ab+1}\)

\(\ge\left(a+b+1\right)\cdot\frac{4}{a+b+2}+\frac{1}{ab+1}\)

\(\ge\frac{4\left(a+b+1\right)}{a+b+2}+\frac{1}{\frac{\left(a+b\right)^2}{4}+1}\)

Đặt \(t=a+b\ge2\sqrt{ab}\ge2\)

Ta cần chứng minh \(\frac{4\left(t+1\right)}{t+2}+\frac{1}{\frac{t^2}{4}+1}\ge\frac{7}{2}\)

\(\Leftrightarrow\frac{\left(t-2\right)^3}{2\left(t+2\right)\left(t^2+4\right)}\ge0\) ( đúng )

Vậy ta có đpcm.

ta có:

\(\frac{xz}{y^2+yz}+\frac{y^2}{xz+yz}+\frac{z+2z}{z+x}=\frac{\frac{xz}{yz}}{\frac{y^2}{yz}+1}+\frac{\frac{y^2}{yz}}{\frac{xz}{yz}+1}+\frac{1+\frac{2z}{x}}{1+\frac{z}{x}}\)\(=\frac{\frac{x}{y}}{\frac{y}{z}+1}+\frac{\frac{y}{z}}{\frac{x}{y}+1}+\frac{1+\frac{2z}{x}}{1+\frac{z}{x}}=\frac{a^2}{b^2+1}+\frac{b^2}{a^2+1}+\frac{1+2c^2}{1+c^2}\)

trong đó \(a^2=\frac{x}{y};b^2=\frac{y}{z};c^2=\frac{z}{x}\left(a;b;c>0\right)\)

Nhận xét rằng \(a^2\cdot b^2=\frac{x}{z}=\frac{1}{c^2}\ge1\)(do x>=z)

Xét \(\frac{a^2}{b^2+1}+\frac{b^2}{a^2+1}+\frac{c^2}{ab+1}\)\(=\frac{a^2\left(a^2+1\right)\left(ab+1\right)+b^2\left(b^2+1\right)\left(ab+1\right)-2aba^2\left(a^2+1\right)\left(b^2+1\right)}{\left(a^2+1\right)\left(b^2+1\right)\left(ab+1\right)}\)

\(=\frac{ab\left(a^2-b^2\right)+\left(a-b\right)\left(a^3-b^3\right)+\left(a-b\right)^2}{\left(a^2+1\right)\left(b^2+1\right)\left(ab+1\right)}\ge0\)

Do đó: \(\frac{a^2}{b^2+1}+\frac{b^2}{a^2+1}\ge\frac{2ab}{ab+1}=\frac{\frac{2}{c}}{\frac{1}{c}+1}=\frac{2}{1+c}\left(1\right)\)đẳng thức xảy ra <=> a=b

khi đó:

\(\frac{2}{1+c}+\frac{1+2c^2}{c^2+1}-\frac{5}{2}=\frac{2\left[2\left(1+c^2\right)+\left(1+c\right)\left(1+2c^2\right)\right]-5\left(1+c\right)\left(1+c^2\right)}{2\left(1+c\right)\left(1+c^2\right)}\)

\(=\frac{1-3c+3c^2-c^3}{2\left(1+c\right)\left(1+c^2\right)}=\frac{\left(1-c\right)^3}{2\left(1+c\right)\left(1+c^2\right)}\ge0\)(do c=<1) (2)

Từ (1) và (2) => đpcm

Đẳng thức xảy ra <=> a=b, c=1 <=> x=y=z

Giờ mình lội lại noti mới nhìn thấy bài tag. Không biết bạn còn cần không nhưng mình vẫn post lời giải để mọi người tham khảo:

Ta có:

\(\frac{xz}{y^2+yz}+\frac{y^2}{xz+yz}+\frac{x+2z}{x+z}=\frac{xz}{y^2+yz}+\frac{y^2}{xz+yz}+\frac{z}{x+z}+1\)

\(=\frac{1}{\frac{y^2}{xz}+\frac{y}{x}}+\frac{1}{\frac{xz}{y^2}+\frac{z}{y}}+\frac{1}{\frac{x}{z}+1}+1\)

Đặt \((\frac{x}{y}, \frac{y}{z})=(a,b)\Rightarrow ab=\frac{x}{z}\geq 1\) do $x\ge z$

BĐT cần CM trở thành:

\(\frac{1}{\frac{b}{a}+\frac{1}{a}}+\frac{1}{\frac{a}{b}+\frac{1}{b}}+\frac{1}{ab+1}+1\geq \frac{5}{2}\)

\(\Leftrightarrow \frac{a}{b+1}+\frac{b}{a+1}+\frac{1}{ab+1}+1\geq \frac{5}{2}\)

\(\Leftrightarrow \frac{a+b+1}{b+1}+\frac{b+a+1}{a+1}+\frac{1}{ab+1}\geq \frac{7}{2}(*)\)

Đăt \(P=\frac{a+b+1}{b+1}+\frac{b+a+1}{a+1}+\frac{1}{ab+1}\). Áp dụng BĐT Cauchy-Schwarz và AM-GM ta có:

\(P\geq (a+b+1).\frac{4}{b+1+a+1}+\frac{1}{(\frac{a+b}{2})^2+1}=\frac{4(a+b+1)}{a+b+2}+\frac{4}{(a+b)^2+4}(1)\)

Đặt \(t=a+b\). Theo BĐT AM-GM \(t=a+b\geq 2\sqrt{ab}\geq 2\sqrt{1}=2\)

Xét hiệu:

\(\frac{4(a+b+1)}{a+b+2}+\frac{4}{(a+b)^2+4}-\frac{7}{2}=\frac{4(t+1)}{t+2}+\frac{4}{t^2+4}-\frac{7}{2}\)

\(=\frac{t^3-6t^2+12t-8}{2(t+2)(t^2+4)}=\frac{(t-2)^3}{2(t+2)(t^2+4)}\geq 0, \forall t\geq 2\)

\(\Rightarrow \frac{4(a+b+1)}{a+b+2}+\frac{4}{(a+b)^2+4}\geq \frac{7}{2}(2)\)

Từ \((1);(2)\Rightarrow P\geq \frac{7}{2}\). BĐT $(*)$ đúng, ta có đpcm.

Dấu "=" xảy ra khi $x=y=z$

Nguyễn Thành Trương Ngân Vũ ThịAkai Haruma

Đk: $x\geq \frac{1}{2}$

Pt $\Leftrightarrow 4x^2+3x-7=4(\sqrt{x^3+3x^2}-2)+2(\sqrt{2x-1}-1)$

$\Leftrightarrow +4\frac{(x-1)(x+2)^2}{\sqrt{x^3+3x^2}+2}+4\frac{x-1}{\sqrt{2x-1}+1}-(x-1)(4x+7)=0$

$\Leftrightarrow (x-1)[\frac{4(x+2)^2}{\sqrt{x^3+3x^2}+2}+\frac{4}{\sqrt{2x-1}+1}-(4x+7)]=0$

$\Leftrightarrow x=1\vee \frac{4(x+2)^2}{\sqrt{x^3+3x^2}+2}+\frac{4}{\sqrt{2x-1}+1}-4x-7=0$ $(*)$

Xét hàm số $f(x)=\frac{4(x+2)^2}{\sqrt{x^3+3x^2}+2}+\frac{4}{\sqrt{2x-1}+1}-4x-7,x\in [\frac{1}{2};+\infty )$ thì $f(x)>0,\forall x\in [\frac{1}{2};+\infty )$

$\Rightarrow $ Pt $(*)$ vô nghiệm

1 slot tối làm cho :))

Bài này trích trong đề thi HSG Toán 9 tỉnh Thanh Hóa

Như đã hứa,giờ làm cho :))

BĐT\(\Leftrightarrow\frac{xz}{y^2+yz}+\frac{y}{xz+yz}+\frac{z}{x+z}\ge\frac{3}{2}\).Đặt \(\frac{x}{y}=a>0;\frac{y}{z}=b>0\)\(\Rightarrow ab=\frac{x}{z}\ge1\)

Ta có BĐT:\(\frac{1}{\frac{y^2}{xz}+\frac{y}{x}}+\frac{1}{\frac{xz}{y^2}+\frac{z}{y}}+\frac{1}{1+\frac{x}{z}}\ge\frac{3}{2}\)

\(\Rightarrow\frac{1}{\frac{b}{a}+\frac{1}{a}}+\frac{1}{\frac{a}{b}+\frac{1}{b}}+\frac{1}{ab+1}\ge\frac{3}{2}\)\(\Leftrightarrow\frac{a}{b+1}+\frac{b}{a+1}+\frac{1}{ab+1}\ge\frac{3}{2}\)

\(\Leftrightarrow\frac{a^2}{ab+a}+\frac{b^2}{ab+b}+\frac{1}{ab+1}\ge\frac{3}{2}\).Áp dụng BĐT Bunhiacopxki mở rộng ta có:

\(\frac{a^2}{ab+a}+\frac{b^2}{ab+b}\ge\frac{\left(a+b\right)^2}{2ab+a+b}\).Ta cần chứng minh:\(\frac{\left(a+b\right)^2}{2ab+a+b}\ge\frac{2\left(a+b\right)}{a+b+2}\)(*).Thật vậy:

(*)\(\Rightarrow\frac{a+b}{2ab+a+b}\ge\frac{2}{a+b+2}\Leftrightarrow\left(a+b\right)\left(a+b+2\right)\ge2\left(2ab+a+b\right)\)

\(\Leftrightarrow a^2-2ab+b^2\ge0\Leftrightarrow\left(a-b\right)^2\ge0\)(đúng)

Nên \(\frac{a^2}{ab+a}+\frac{b^2}{ab+b}+\frac{1}{ab+1}\ge\frac{2\left(a+b\right)}{a+b+2}+\frac{1}{ab+1}\)\(\ge\frac{2\left(a+b\right)}{a+b+2}+\frac{4}{4+\left(a+b\right)^2}\)

Đặt \(m=a+b\ge2\sqrt{ab}\ge2\).Ta cần chứng minh:\(\frac{2m}{m+2}+\frac{4}{4+m^2}\ge\frac{3}{2}\)(**).Thật vậy

(**)\(\Leftrightarrow\frac{2m}{m+2}+\frac{3m^2+4}{2m^2+8}\ge0\)\(\Leftrightarrow\frac{2m\left(2m^2+8\right)-\left(m+2\right)\left(3m^2+4\right)}{\left(m+2\right)\left(2m^2+8\right)}\ge0\)

\(\Leftrightarrow\frac{\left(m-2\right)^3}{\left(m+2\right)\left(2m^2+8\right)}\ge0\) đúng với mọi \(m\ge2\)

Vậy BĐT đã được chứng minh.Dấu "=" xảy ra khi chỉ khi x=y=z

Xét \(VT=\frac{xz}{y^2+yz}+\frac{y^2}{xz+yz}+\frac{x+2z}{x+z}\)

\(=\frac{\frac{x}{y}}{\frac{y}{z}+1}+\frac{\frac{y}{z}}{\frac{x}{y}+1}+1+\frac{1}{\frac{x}{z}+1}\)

Đặt \(\frac{x}{y}=u,\frac{y}{z}=v\left(u,v>0\right)\Rightarrow\frac{x}{z}=uv\ge1\)(Do \(x\ge z\))

Khi đó vế trái được viết lại thành: \(\frac{u}{v+1}+\frac{v}{u+1}+1+\frac{1}{uv+1}\ge\frac{5}{2}\)

\(\Leftrightarrow\frac{u}{v+1}+\frac{v}{u+1}+\frac{1}{uv+1}\ge\frac{3}{2}\)với \(uv\ge1\)

Theo BĐT Bunhiacopxki dạng phân thức, ta có: \(\frac{u}{v+1}+\frac{v}{u+1}=\frac{u^2}{uv+u}+\frac{v^2}{uv+v}\ge\frac{\left(u+v\right)^2}{2uv+u+v}\)

\(\ge\frac{\left(u+v\right)^2}{\left(u+v\right)+\frac{\left(u+v\right)^2}{2}}=\frac{2\left(u+v\right)}{u+v+2}\)

Mặt khác: \(\frac{1}{uv+1}\ge\frac{1}{\frac{\left(u+v\right)^2}{4}+1}=\frac{4}{\left(u+v\right)^2+4}\)

Khi đó ta quy BĐT cần chứng minh về: \(\frac{2\left(u+v\right)}{u+v+2}+\frac{4}{\left(u+v\right)^2+4}\ge\frac{3}{2}\)(*)

Đặt \(w=u+v\ge2\sqrt{uv}\ge2\). Khi đó (*) trở thành \(\frac{2w}{w+2}+\frac{4}{w^2+4}\ge\frac{3}{2}\)với \(w\ge2\)

\(\Leftrightarrow\frac{\left(w-2\right)^2}{2\left(w+2\right)\left(w^2+4\right)}\ge0\)(đúng với mọi \(w\ge2\))

Đẳng thức xảy ra khi \(\hept{\begin{cases}u+v=2\\uv=1\\u=v\end{cases}}\Leftrightarrow u=v=1\)hay x = y = z

Bạn tham khảo câu trả lời của mình và các bạn tại đây:

Câu hỏi của Lê Thành An - Toán lớp 9 - Học toán với OnlineMath

https://olm.vn/hoi-dap/detail/253622963565.html ( link nếu bạn ngại vào TKHĐ )

\(VT=\frac{x^2}{x^3-xyz-2013x}+\frac{y^2}{y^3-xyz-2013y}+\frac{z^2}{z^3-xyz-2013z}\ge\frac{\left(x+y+z\right)^2}{x^3+y^3+z^3-3xyz-2013\left(x+y+z\right)}\)

\(=\frac{\left(x+y+z\right)^2}{x^3+y^3+z^3+3\left[\left(x+y+z\right)\left(xy+yz+zx\right)-xyz\right]}=\frac{\left(x+y+z\right)^2}{\left(x+y+z\right)^3}=\frac{1}{x+y+z}\)=VP

đúng rồi ạ nhưng chỉ cần c/m đẳng thức phụ như thế này thôi ạ\(\frac{a^2}{x}+\frac{b^2}{y}\ge\frac{\left(a+b\right)^2}{x+y}\) =>\(\frac{\left(a+b\right)2}{x+y}+\frac{c^2}{z}\ge\frac{\left(a+b+c\right)^2}{x+y+z}\) hay \(\frac{a^2}{x}+\frac{b^2}{y}+\frac{c^2}{z}\ge\frac{\left(a+b+c\right)^2}{x+y+z}\) là xong

\(3=x^2+y^2+z^2\ge3\sqrt[3]{x^2y^2z^2}\)

\(\Rightarrow xyz\le1\)

\(\sqrt[3]{x^2}+\sqrt[3]{y^2}+\sqrt[3]{z^2}\le\frac{x^2+1+1}{3}+\frac{y^2+1+1}{3}+\frac{z^2+1+1}{3}=3\)

Ta co:

\(A=\frac{x}{\sqrt[3]{yz}}+\frac{y}{\sqrt[3]{xz}}+\frac{z}{\sqrt[3]{xy}}=\frac{x\sqrt[3]{x}}{\sqrt[3]{xyz}}+\frac{y\sqrt[3]{y}}{\sqrt[3]{xyz}}+\frac{z\sqrt[3]{z}}{\sqrt[3]{xyz}}\)

\(\ge x\sqrt[3]{x}+y\sqrt[3]{y}+z\sqrt[3]{z}\)

\(\Rightarrow3A\ge3\left(x\sqrt[3]{x}+y\sqrt[3]{y}+z\sqrt[3]{z}\right)\ge\left(x\sqrt[3]{x}+y\sqrt[3]{y}+z\sqrt[3]{z}\right)\left(\sqrt[3]{x^2}+\sqrt[3]{y^2}+\sqrt[3]{z^2}\right)\)

\(\ge\left(x+y+z\right)^2\ge3\left(xy+yz+zx\right)\)

\(\Rightarrow A\ge xy+yz+zx\)

Áp dụng BĐT Cauchy - Schwarz, ta có: \(3\left(x^2+y^2+z^2\right)=\left(1^2+1^2+1^2\right)\left(x^2+y^2+z^2\right)\ge\left(x+y+z\right)^2\)

\(\Rightarrow x+y+z\le\sqrt{3\left(x^2+y^2+z^2\right)}=3=x^2+y^2+z^2\)(Do \(x^2+y^2+z^2=3\))

Ta có: \(\frac{x}{\sqrt[3]{yz}}+\frac{y}{\sqrt[3]{zx}}+\frac{z}{\sqrt[3]{xy}}=\frac{x}{\sqrt[3]{yz.1}}+\frac{y}{\sqrt[3]{zx.1}}+\frac{z}{\sqrt[3]{xy.1}}\)

\(\ge\frac{x}{\frac{y+z+1}{3}}+\frac{y}{\frac{z+x+1}{3}}+\frac{z}{\frac{x+y+1}{3}}\)\(=\frac{3x}{y+z+1}+\frac{3y}{z+x+1}+\frac{3z}{x+y+1}\)

\(=\frac{3x^2}{xy+zx+x}+\frac{3y^2}{yz+xy+y}+\frac{3z^2}{zx+yz+z}\)\(\ge\frac{3\left(x+y+z\right)^2}{2\left(xy+yz+zx\right)+\left(x+y+z\right)}\)(Theo BĐT Cauchy - Schwarz dạng Engle)

\(\ge\frac{3\left(x+y+z\right)^2}{2\left(xy+yz+zx\right)+x^2+y^2+z^2}=\frac{3\left(x+y+z\right)^2}{\left(x+y+z\right)^2}=3=x^2+y^2+z^2\)

\(\ge xy+yz+zx\)

Đẳng thức xảy ra khi x = y = z = 1