\(\sqrt{x^2+2024}=\sqrt{x^2+xy+yz+zx}=\sqrt{\left(x+y\right)\left(z+x\right)}\ge\sqrt{\left(\sqrt{xz}+\sqrt{xy}\right)^2}=\sqrt{xy}+\sqrt{xz}\)

Tương tự: \(\sqrt{y^2+2024}\ge\sqrt{xy}+\sqrt{yz}\)

\(\sqrt{z^2+2024}\ge\sqrt{xz}+\sqrt{yz}\)

Cộng vế:

\(P\ge\dfrac{2\left(\sqrt{xy}+\sqrt{yz}+\sqrt{zx}\right)}{\sqrt{xy}+\sqrt{yz}+\sqrt{zx}}=2\)

Dấu "=" xảy ra khi \(x=y=z=\dfrac{2024}{3}\)

Nhìn qua thấy bậc của bđt là không đồng bậc nên hơi căng đấy...

Chú ý: \(2019=\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}=\frac{x+y+z}{xyz}\Rightarrow xyz=\frac{x+y+z}{2019}\)

\(LHS=\Sigma_{cyc}\frac{\sqrt{2019x^2+1}+1}{x}=\Sigma_{cyc}\frac{\sqrt{\frac{x}{y}+\frac{x^2}{yz}+\frac{x}{z}+1}+1}{x}\)( thay \(2019=\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}\))

\(=\Sigma_{cyc}\frac{\sqrt{\left(\frac{x}{y}+1\right)\left(\frac{x}{z}+1\right)}+1}{x}=\Sigma_{cyc}\left[\sqrt{\frac{\left(\frac{x}{y}+1\right)}{x}.\frac{\left(\frac{x}{z}+1\right)}{x}}+\frac{1}{x}\right]\)

\(=\Sigma_{cyc}\sqrt{\left(\frac{1}{y}+\frac{1}{x}\right)\left(\frac{1}{z}+\frac{1}{x}\right)}+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\le\frac{1}{2}\left[4\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\right]+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\)

\(=3\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)=\frac{3\left(xy+yz+zx\right)}{xyz}=\frac{3\left(xy+yz+zx\right)}{\frac{\left(x+y+z\right)}{2019}}=\frac{6057\left(xy+yz+zx\right)}{x+y+z}\)

\(\le\frac{6057.\frac{\left(x+y+z\right)^2}{3}}{x+y+z}=2019\left(x+y+z\right)\)(đpcm)

Đẳng thức xảy ra khi \(x=y=z=\sqrt{\frac{3}{2019}}\)

P/s: Check hộ t phát:3

Đặt \(a=\frac{1}{x};b=\frac{1}{y};c=\frac{1}{z}\)thì bài toán thành

Cho: \(ab+bc+ca=2019\)

Chứng minh:

\(\sqrt{2019+a^2}+\sqrt{2019+b^2}+\sqrt{2019+c^2}+\left(a+b+c\right)\le2019\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)

Ta có:

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

\(VT=\sqrt{\left(a+b\right)\left(a+c\right)}+\sqrt{\left(b+a\right)\left(b+c\right)}+\sqrt{\left(c+a\right)\left(c+b\right)}+\left(a+b+c\right)\)

\(\le3\left(a+b+c\right)\)

\(VP=\left(ab+bc+ca\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)

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

\(\ge3\left(a+b+c\right)\)

Tới đây bí :(

\(A=\dfrac{xyz.x}{xy+xyz.x+xyz}+\dfrac{y}{yz+y+2019}+\dfrac{yz}{xyz+yz+y}\)

\(=\dfrac{xz}{1+xz+z}+\dfrac{y}{yz+y+2019}+\dfrac{yz}{yz+y+2019}\)

\(=\dfrac{xyz}{y+xyz+yz}+\dfrac{y}{yz+y+2019}+\dfrac{yz}{yz+y+2019}\)

\(=\dfrac{2019}{y+2019+yz}+\dfrac{y}{yz+y+2019}+\dfrac{yz}{yz+y+2019}\)

\(=\dfrac{yz+y+2019}{yz+y+2019}=1\)

\(VT=\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=\dfrac{1}{2}\left(\dfrac{1}{x}+\dfrac{1}{y}\right)+\dfrac{1}{2}\left(\dfrac{1}{y}+\dfrac{1}{z}\right)+\dfrac{1}{2}\left(\dfrac{1}{z}+\dfrac{1}{x}\right)\ge\dfrac{1}{2}.\dfrac{2}{\sqrt{xy}}+\dfrac{1}{2}.\dfrac{1}{\sqrt{yz}}+\dfrac{1}{2}.\dfrac{1}{\sqrt{zx}}=\dfrac{1}{\sqrt{xy}}+\dfrac{1}{\sqrt{yz}}+\dfrac{1}{\sqrt{zx}}\)

Dấu "=" xảy ra <=> x = y = z > 0

Bài 1:
Vì $x+y+z=1$ nên:

\(Q=\frac{x}{x+\sqrt{x(x+y+z)+yz}}+\frac{y}{y+\sqrt{y(x+y+z)+xz}}+\frac{z}{z+\sqrt{z(x+y+z)+xy}}\)

\(Q=\frac{x}{x+\sqrt{(x+y)(x+z)}}+\frac{y}{y+\sqrt{(y+z)(y+x)}}+\frac{z}{z+\sqrt{(z+x)(z+y)}}\)

Áp dụng BĐT Bunhiacopxky:

\(\sqrt{(x+y)(x+z)}=\sqrt{(x+y)(z+x)}\geq \sqrt{(\sqrt{xz}+\sqrt{xy})^2}=\sqrt{xz}+\sqrt{xy}\)

\(\Rightarrow \frac{x}{x+\sqrt{(x+y)(x+z)}}\leq \frac{x}{x+\sqrt{xy}+\sqrt{xz}}=\frac{\sqrt{x}}{\sqrt{x}+\sqrt{y}+\sqrt{z}}\)

Hoàn toàn tương tự với các phân thức còn lại và cộng theo vế suy ra:

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

Vậy $Q$ max bằng $1$

Dấu bằng xảy ra khi $x=y=z=\frac{1}{3}$

Bài 2:
Vì $x+y+z=1$ nên:

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

\(\text{VT}=\frac{(x+y+z)^2-x^2}{(x+y)(x+z)}+\frac{(x+y+z)^2-y^2}{(y+z)(y+x)}+\frac{(x+y+z)^2-z^2}{(z+x)(z+y)}\)

\(\text{VT}=\frac{(y+z)[(x+y)+(x+z)]}{(x+y)(x+z)}+\frac{(x+z)[(y+z)+(y+x)]}{(y+z)(y+x)}+\frac{(x+y)[(z+x)+(z+y)]}{(z+x)(z+y)}\)

Áp dụng BĐT AM-GM:
\(\text{VT}\geq \frac{2(y+z)\sqrt{(x+y)(x+z)}}{(x+y)(x+z)}+\frac{2(x+z)\sqrt{(y+z)(y+x)}}{(y+z)(y+x)}+\frac{2(x+y)\sqrt{(z+x)(z+y)}}{(z+x)(z+y)}\)

\(\Leftrightarrow \text{VT}\geq 2\underbrace{\left(\frac{y+z}{\sqrt{(x+y)(x+z)}}+\frac{x+z}{\sqrt{(y+z)(y+x)}}+\frac{x+y}{\sqrt{(z+x)(z+y)}}\right)}_{M}\)

Tiếp tục AM-GM cho 3 số trong ngoặc lớn, suy ra \(M\geq 3\)

Do đó: \(\text{VT}\geq 2.3=6\) (đpcm)

Dấu bằng xảy ra khi $3x=3y=3z=1$

\(M=\dfrac{yz\sqrt{x-1}+xz\sqrt{y-2}+xy\sqrt{z-3}}{xyz}\)

\(=\dfrac{yz\sqrt{x-1}}{xyz}+\dfrac{xz\sqrt{y-2}}{xyz}+\dfrac{xy\sqrt{z-3}}{xyz}\)

\(=\dfrac{\sqrt{x-1}}{x}+\dfrac{\sqrt{y-2}}{y}+\dfrac{\sqrt{z-3}}{z}\)

Áp dụng BĐT AM-GM ta có:

\(\sqrt{x-1}\le\dfrac{1+x-1}{2}=\dfrac{x}{2}\)\(\Rightarrow\dfrac{\sqrt{x-1}}{x}\le\dfrac{x}{2}\cdot\dfrac{1}{x}=\dfrac{1}{2}\)

\(\sqrt{y-2}=\dfrac{\sqrt{2\left(y-2\right)}}{\sqrt{2}}\le\dfrac{y}{2\sqrt{2}}\)\(\Rightarrow\dfrac{\sqrt{y-2}}{y}\le\dfrac{y}{2\sqrt{2}}\cdot\dfrac{1}{y}=\dfrac{1}{2\sqrt{2}}\)

\(\sqrt{z-3}=\dfrac{\sqrt{3\left(z-3\right)}}{\sqrt{3}}\le\dfrac{z}{2\sqrt{3}}\)\(\Rightarrow\dfrac{\sqrt{z-3}}{z}\le\dfrac{z}{2\sqrt{3}}\cdot\dfrac{1}{z}=\dfrac{1}{2\sqrt{3}}\)

Cộng theo vế 3 BĐT trên ta có:

\(M\le\dfrac{1}{2}\left(1+\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}\right)\) (ĐPCM)

Ta có: \(P=\frac{\sqrt{yz}}{x+2\sqrt{yz}}+\frac{\sqrt{zx}}{y+2\sqrt{zx}}+\frac{\sqrt{xy}}{z+2\sqrt{xy}}=\frac{1}{\frac{x}{\sqrt{yz}}+2}+\frac{1}{\frac{y}{\sqrt{zx}}+2}+\frac{1}{\frac{z}{\sqrt{xy}}+2}\)

Đặt \(\frac{x}{\sqrt{yz}}=c,\frac{y}{\sqrt{zx}}=t;\frac{z}{\sqrt{xy}}=k\left(c,t,k>0\right)\)thì ctk = 1

Ta cần tìm giá trị lớn nhất của \(P=\frac{1}{c+2}+\frac{1}{t+2}+\frac{1}{k+2}\)với ctk = 1

Dự đoán MaxP = 1 khi c = t = k = 1

Thật vậy: \(P=\frac{kt+2k+2t+4+ct+2c+2t+4+ck+2c+2k+4}{\left(c+2\right)\left(t+2\right)\left(k+2\right)}=\frac{\left(kt+tc+ck\right)+4\left(c+t+k\right)+12}{ctk+2\left(kt+tc+ck\right)+4\left(c+t+k\right)+8}\le\frac{\left(kt+tc+ck\right)+4\left(c+t+k\right)+12}{1+\left(kt+tc+ck\right)+3\sqrt[3]{\left(ctk\right)^2}+4\left(c+t+k\right)+8}=1\)Đẳng thức xảy ra khi x = y = z

Ta có: \(\frac{\sqrt{yz}}{x+2\sqrt{yz}}=\frac{1}{2}\left(1-\frac{x}{x+2\sqrt{yz}}\right)\le\frac{1}{2}\left(1-\frac{x}{x+y+z}\right)=\frac{1}{2}\left(\frac{y+z}{x+y+z}\right)\)(bđt cosi) (1)

CMTT: \(\frac{\sqrt{xz}}{y+2\sqrt{xz}}\le\frac{1}{2}\left(\frac{x+z}{x+y+z}\right)\)(2)

\(\frac{\sqrt{xy}}{z+2\sqrt{xy}}\le\frac{1}{2}\left(\frac{x+y}{x+y+z}\right)\)(3)

Từ (1), (2) và (3) cộng vế theo vế ta có:

\(\frac{\sqrt{yz}}{x+2\sqrt{yz}}+\frac{\sqrt{xz}}{y+2\sqrt{xz}}+\frac{\sqrt{xy}}{z+2\sqrt{xy}}\le\frac{1}{2}\left(\frac{y+z}{x+y+z}\right)+\frac{1}{2}\left(\frac{x+z}{x+y+z}\right)+\frac{1}{2}\left(\frac{x+y}{x+y+z}\right)\)

=> P \(\le\frac{1}{2}\left(\frac{y+z+x+z+x+y}{x+y+z}\right)=\frac{1}{2}\cdot\frac{2\left(x+y+z\right)}{x+y+z}=1\)

Dấu "=" xảy ra <=> x = y = z

Vậy MaxP = 1 <=> x = y = z