\(A=\left(1+\frac{y}{2x}+2x+y\right)\left(1+\frac{4}{\sqrt{y}}\right)^2\ge\left(1+2\sqrt{y}+y\right)\left(1+\frac{4}{\sqrt{y}}\right)^2\)
\(\Rightarrow A\ge\left(1+\sqrt{y}\right)^2\left(1+\frac{4}{\sqrt{y}}\right)^2=\left(1+\frac{4}{\sqrt{y}}+\sqrt{y}+4\right)^2\ge\left(1+2\sqrt{4}+4\right)^2=81\)
Dấu "=" xảy ra khi \(\left\{{}\begin{matrix}x=1\\y=4\end{matrix}\right.\)
Cho x,y>0 Chứng minh rằng:\(\left(x+1\right)\left(1+\frac{y}{x}\right)\left(1+\frac{9}{\sqrt{y}}\right)^2\ge256\)
1) Giaỉ hệ \(\left\{{}\begin{matrix}\left(x+y\right)^2\left(8x^2+8y^2+4xy-13\right)+5=0\\2x+\frac{1}{x+y}=1\end{matrix}\right.\)
2) Tính P\(=\frac{4\left(x+1\right)x^{2018}-2x^{2017}+2x+1}{2x^2+3x}\)Với x\(=\sqrt{\frac{1}{2\sqrt{3}-2}-\frac{3}{2\sqrt{3}+2}}\)
1. gpt : \(\frac{2x+1}{\sqrt{x^2+2}}+\left(x+1\right)\sqrt{1+\frac{2x+1}{x^2+2}}+x=0\)
2. \(\left\{{}\begin{matrix}x,y,z>0\\x+y+z\le\frac{3}{2}\end{matrix}\right.\) Tìm min \(Q=\frac{x}{y^2z}+\frac{y}{z^2x}+\frac{z}{x^2y}+\frac{x^5}{y}+\frac{y^5}{z}+\frac{z^5}{x}\)
Giải PT và HPT:
1)\(\left\{{}\begin{matrix}xy+x+y=3\\\frac{1}{x^2+2x}+\frac{1}{y^2+2y}=\frac{2}{3}\end{matrix}\right.\)
2)\(\left(\sqrt{x+4}-2\right)\left(\sqrt{4-x}+2\right)=2x\)
3)\(\left\{{}\begin{matrix}xy\left(x+y\right)=2\\9xy\left(3x-y\right)+6=26x^3-2y^3\end{matrix}\right.\)
4)\(\left\{{}\begin{matrix}x^2-2xy+x-2y+3=0\\y^2-x^2+2xy+2x-2=0\end{matrix}\right.\)
giải hệ phương trình:
1, \(\left\{{}\begin{matrix}2y\left(4y^2+3x^2\right)=x^4\left(x^2+3\right)\\2012^x\left(\sqrt{2y-2x+5}-x+1\right)=4024\end{matrix}\right.\)
2, \(\left\{{}\begin{matrix}x^3-2x^2y-15x=6y\left(2x-5-4y\right)\\\frac{x^2}{8y}+\frac{2x}{3}=\sqrt{\frac{x^3}{3y}+\frac{x^2}{4}}-\frac{y}{2}\end{matrix}\right.\)
3, \(\left\{{}\begin{matrix}8\left(x^2+y^2\right)+4xy+\frac{5}{\left(x+y\right)^2}=13\\2x+\frac{1}{x+y}=1\end{matrix}\right.\)
Cho các số thực x,y thỏa mãn \(\left(x+1+\sqrt{x^2+2x+2}\right)\left(y-1+\sqrt{y^2-2y+2}\right)\)=0 ,chứng minh rằng x+y=0
Giải hệ phương trình :
1, \(\left\{{}\begin{matrix}\frac{2}{x}+\frac{3}{y-2}=4\\\frac{4}{x}+\frac{1}{y-2}=1\end{matrix}\right.\)
2 , \(\left\{{}\begin{matrix}\frac{2}{2x-y}-\frac{1}{x+y}=0\\\frac{3}{2x-y}-\frac{6}{x+y}=-1\end{matrix}\right.\)
3, \(\left\{{}\begin{matrix}5\left(x+2y\right)=3x-1\\2x+4=3\left(x-2y\right)-15\end{matrix}\right.\)
4, \(\left\{{}\begin{matrix}2x+y=7\\-x+4y=10\end{matrix}\right.\)
Giải hệ:
\(\left\{{}\begin{matrix}x^2+y^2+xy=5\\27x^3+6y^2x=2+y^3+30x^2y\end{matrix}\right.\)
\(\left\{{}\begin{matrix}x^2+y^2+\frac{8xy}{x+y}=16\\\frac{x^2}{8y}+\frac{2x}{3}=\sqrt{\frac{x^3}{3y}+\frac{x^2}{4}}-\frac{y}{2}\end{matrix}\right.\), \(\left\{{}\begin{matrix}\frac{1}{3x}+\frac{2x}{3y}=\frac{x+\sqrt{y}}{2x^2+y}\\2\left(2x+\sqrt{y}\right)=\sqrt{2x+6}-y\end{matrix}\right.\)
\(\left\{{}\begin{matrix}x^2y-3x-1=3x\sqrt{y}\left(\sqrt{1-x}-1\right)^3\\\sqrt{8x^2-3xy+4y^2}+\sqrt{xy}=4y\end{matrix}\right.\)
Cho các số a,b,c là các số k âm sao cho tổng hai số bất kì đều dương.CMR \(\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}+\frac{16\sqrt{ab+bc+ac}}{a+b+c}\ge8\)
1) Giaỉ hệ phương trình \(\left\{{}\begin{matrix}x^2-2xy+x-2y+3=0\\y^2-x^2+2xy+2x-2=0\end{matrix}\right.\)
2) Giaỉ hệ phương trình \(\left\{{}\begin{matrix}x^2-6y^2-xy-2x+11y=3\\x^2+y^2=5\end{matrix}\right.\)
3) Chứng minh biểu thức sau không là số tự nhiên S= \(1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+....+\frac{1}{\sqrt{100}}\)
