Do \(a^4+b^6+c^8\le1\Rightarrow\left|a\right|;\left|b\right|;\left|c\right|\le1\)
\(\Rightarrow a^4\ge\left|a\right|^{2001}\ge a^{2001}\)
Tương tự \(b^6\ge b^{2003}\) ; \(c^8\ge c^{2003}\)
\(\Rightarrow a^{2001}+b^{2003}+c^{2003}\le1\)
Kết hợp giả thiết \(\Rightarrow a^{2001}+b^{2003}+c^{2003}=1\)
Dấu "=" xảy ra khi \(\left(a;b;c\right)=\left(1;0;0\right)\) và các hoán vị
a)\(\left\{{}\begin{matrix}\frac{x-12}{4}=\frac{y-9}{3}=z-1\\3x+5y-z=2\end{matrix}\right.\)
b)\(\left\{{}\begin{matrix}\frac{a+b}{6}=\frac{b+c}{7}=\frac{a+c}{8}\\a+b+c=14\end{matrix}\right.\)
\(\left\{{}\begin{matrix}\frac{x-12}{4}=\frac{y-9}{3}=z-1\\3x+5y-z=2\end{matrix}\right.\)
\(\left\{{}\begin{matrix}\frac{a+b}{6}=\frac{b+c}{7}=\frac{a+c}{8}\\a+b+c=14\end{matrix}\right.\)
a)\(\left\{{}\begin{matrix}\frac{x-12}{4}=\frac{y-9}{3}=z-1\\3x+5y-z=2\end{matrix}\right.\)
b)\(\left\{{}\begin{matrix}\frac{a+b}{6}=\frac{b+c}{7}\frac{a+c}{8}\\a+b+c=14\end{matrix}\right.\)
c)\(\left\{{}\begin{matrix}x+y+z=9\\\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1\\zy+yz+zx=27\end{matrix}\right.\)
Giải hệ pt sau
a)\(\left\{{}\begin{matrix}\frac{x-12}{4}=\frac{y-9}{3}=z-1\\3x+5y-z=2\end{matrix}\right.\)
b)\(\left\{{}\begin{matrix}\frac{a+b}{6}=\frac{b+c}{7}=\frac{a+c}{8}\\a+b+c=14\end{matrix}\right.\)
1. \(\left\{{}\begin{matrix}a,b,c>0\\a+b+c=1\end{matrix}\right.\). Cmr: \(\frac{ab}{\sqrt{\left(1-c\right)^2\left(1+c\right)}}+\frac{bc}{\sqrt{\left(1-a\right)^2\left(1+a\right)}}+\frac{ca}{\sqrt{\left(1-b\right)^3\left(1+b\right)}}\le\frac{3\sqrt{2}}{8}\)
2. \(\left\{{}\begin{matrix}a,b,c>0\\a+b+c\le1\end{matrix}\right.\). Cmr: \(\frac{1}{a^2+b^2+c^2}+\frac{1}{ab\left(a+b\right)}+\frac{1}{bc\left(b+c\right)}+\frac{1}{ac\left(a+c\right)}\ge\frac{87}{2}\)
3. \(\left\{{}\begin{matrix}a,b,c>0\\ab+bc+ca=2abc\end{matrix}\right.\). Cmr: \(\frac{1}{a\left(2a-1\right)^2}+\frac{1}{b\left(2b-1\right)^2}+\frac{1}{c\left(2c-1\right)^2}\ge\frac{1}{2}\)
4. \(\left\{{}\begin{matrix}x,y,z>0\\x+y+z=2015\end{matrix}\right.\). Tìm min \(A=\frac{x^4+y^4}{x^3+y^3}+\frac{y^4+z^4}{y^3+z^3}+\frac{z^4+x^4}{z^2+x^2}\)
Mn giúp mk với ạ! Thanks nhiều
\(\left\{{}\begin{matrix}a,b,c\ge0\\a+b+c=2\end{matrix}\right.\) Cmr: \(a^2b^2+b^2c^2+c^2a^2-2abc\le1\)
1. giải pt và hpt : a) \(3x-16y-24=\sqrt{9x^2+16x+32}\) (\(x,y\in N\)*)
b) \(4x^3+5x^2+1=\sqrt{3x+1}-3x\)
c) \(\left\{{}\begin{matrix}y^2\sqrt{2x-1}+\sqrt{3}=5y^2-\sqrt{6x-3}\\2y^4\left(5x^2-17x+6\right)=6-15x\end{matrix}\right.\)
2. \(\left\{{}\begin{matrix}a,b,c\ge1\\32abc=18\left(a+b+c\right)+24\end{matrix}\right.\) Tìm Max \(P=\frac{\sqrt{a^2-1}}{a}+\frac{\sqrt{b^2-1}}{b}+\frac{\sqrt{c^2-1}}{c}\)
\(\left\{{}\begin{matrix}a,b,c>0\\abc=1\end{matrix}\right.\) Cmr: \(\frac{1}{2+a}+\frac{1}{2+b}+\frac{1}{2+c}\le1\)
Cho các số thực a,b,c,d thoả mãn:\(\left\{{}\begin{matrix}a+b=c+d\\ab+1=cd\end{matrix}\right.\).CMR:c=d
