A>=1/(1+xy)=1/2

Dấu = xảy ra khi x=y=1

Ta có \(x^2+y^2+xy+x=y-1\)

\(\Leftrightarrow2x^2+2y^2+2xy+2x-2y+2=0\)

\(\Leftrightarrow\left(x+y\right)^2+\left(x+1\right)^2+\left(y-1\right)^2=0\)

\(\Leftrightarrow\left\{{}\begin{matrix}x+y=0\\x+1=0\\y-1=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x=-1\\y=1\end{matrix}\right.\)

\(\Rightarrow B=\left(-1+1-1\right)^{2023}\) \(=\left(-1\right)^{2023}\) \(=-1\)

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Ta có: \(x^2+y^2=\left(x+y\right)^2-2xy=9-2=7\)

\(x^3+y^3=\left(x+y\right)^3-3xy\left(x+y\right)=3^3-3.3=18\)

=> \(x^5+y^5=\left(x^2+y^2\right)\left(x^3+y^3\right)-x^2y^2\left(x+y\right)\)

\(=7.18-1.3=123\)

Lời giải:

$xy+\sqrt{(1+x^2)(1+y^2)}=1$

$\Leftrightarrow \sqrt{(1+x^2)(1+y^2)}=1-xy$

$\Rightarrow (1+x^2)(1+y^2)=(1-xy)^2$ (bp 2 vế)

$\Leftrightarrow x^2+y^2=-2xy$

$\Leftrightarrow (x+y)^2=0\Leftrightarrow x=-y$.

Khi đó:

$M=(x+\sqrt{1+(-x)^2})(-x+\sqrt{1+x^2})=(\sqrt{1+x^2}+x)(\sqrt{1+x^2}-x)$

$=1+x^2-x^2=1$

\(P=\dfrac{x^2+y^2+6}{x+y}=\dfrac{x^2+y^2+2xy+4}{x+y}=\dfrac{\left(x+y\right)^2+4}{x+y}=x+y+\dfrac{4}{x+y}\)

\(P\ge2\sqrt{\left(x+y\right).\dfrac{4}{x+y}}=4\)

\(P_{min}=4\) khi \(x=y=1\)

\(1\ge x+\dfrac{1}{y}\ge2\sqrt{\dfrac{x}{y}}\Rightarrow\dfrac{x}{y}\le\dfrac{1}{4}\)

Đặt \(\dfrac{x}{y}=a\Rightarrow0< a\le\dfrac{1}{4}\)

\(P=\dfrac{\left(\dfrac{x}{y}\right)^2-\dfrac{2x}{y}+2}{\dfrac{x}{y}+1}=\dfrac{a^2-2a+2}{a+1}=\dfrac{4a^2-8a+8}{4\left(a+1\right)}=\dfrac{4a^2-13a+3+5\left(a+1\right)}{4\left(a+1\right)}\)

\(P=\dfrac{5}{4}+\dfrac{\left(1-4a\right)\left(3-a\right)}{4\left(a+1\right)}\ge\dfrac{5}{4}\)

Dấu "=" xảy ra khi \(a=\dfrac{1}{4}\) hay \(\left(x;y\right)=\left(\dfrac{1}{2};2\right)\)

\(x\ge xy+1\Rightarrow1\ge y+\dfrac{1}{x}\ge2\sqrt{\dfrac{y}{x}}\Rightarrow\dfrac{y}{x}\le\dfrac{1}{4}\)

\(Q^2=\dfrac{x^2+2xy+y^2}{3x^2-xy+y^2}=\dfrac{\left(\dfrac{y}{x}\right)^2+2\left(\dfrac{y}{x}\right)+1}{\left(\dfrac{y}{x}\right)^2-\dfrac{y}{x}+3}\)

Đặt \(\dfrac{y}{x}=t\le\dfrac{1}{4}\) 

\(Q^2=\dfrac{t^2+2t+1}{t^2-t+3}=\dfrac{t^2+2t+1}{t^2-t+3}-\dfrac{5}{9}+\dfrac{5}{9}\)

\(Q^2=\dfrac{\left(4t-1\right)\left(t+6\right)}{9\left(t^2-t+3\right)}+\dfrac{5}{9}\le\dfrac{5}{9}\)

\(\Rightarrow Q_{max}=\dfrac{\sqrt{5}}{3}\) khi \(t=\dfrac{1}{4}\) hay \(\left(x;y\right)=\left(2;\dfrac{1}{2}\right)\)