Violympic toán 9
Các câu hỏi tương tự
1) dfrac{x-3x^2}{2}+sqrt{2x^4-x^3+7x^2-3x+3}2
2) 1+sqrt{dfrac{x-2}{1-x}}dfrac{2x^2-2x+1}{x^2-2x+2}
3) x+y+z+dfrac{3}{x-1}+dfrac{3}{y-1}+dfrac{3}{z-1}2left(sqrt{x+2}+sqrt{y+2}+sqrt{z+2}right) với x ,y ,z 1
4) sqrt[3]{x+6}+x^27-sqrt{x-1}
5) x^4-2x^3+x-sqrt{2left(x^2-xright)}0
Đọc tiếp
1) \(\dfrac{x-3x^2}{2}+\sqrt{2x^4-x^3+7x^2-3x+3}=2\)
2) \(1+\sqrt{\dfrac{x-2}{1-x}}=\dfrac{2x^2-2x+1}{x^2-2x+2}\)
3) \(x+y+z+\dfrac{3}{x-1}+\dfrac{3}{y-1}+\dfrac{3}{z-1}=2\left(\sqrt{x+2}+\sqrt{y+2}+\sqrt{z+2}\right)\) với x ,y ,z > 1
4) \(\sqrt[3]{x+6}+x^2=7-\sqrt{x-1}\)
5) \(x^4-2x^3+x-\sqrt{2\left(x^2-x\right)}=0\)
Cho x, y, z 0 thoả mãn x+y+z2. Tìm GTNN của các biểu thức:
a) Asqrt{x^2+dfrac{1}{x^2}}+sqrt{y^2+dfrac{1}{y^2}}+sqrt{z^2+dfrac{1}{z^2}}
b) Bsqrt{x^2+dfrac{1}{y^2}+dfrac{1}{z^2}}+sqrt{y^2+dfrac{1}{z^2}+dfrac{1}{x^2}}+sqrt{z^2+dfrac{1}{x^2}+dfrac{1}{y^2}}
c) Csqrt{2x^2+dfrac{3}{y^2}+dfrac{4}{z}}+sqrt{2y^2+dfrac{3}{z^2}+dfrac{4}{x^2}}+sqrt{2z^2+dfrac{3}{x^2}+dfrac{4}{y^2}}
Đọc tiếp
Cho x, y, z > 0 thoả mãn x+y+z=2. Tìm GTNN của các biểu thức:
a) \(A=\sqrt{x^2+\dfrac{1}{x^2}}+\sqrt{y^2+\dfrac{1}{y^2}}+\sqrt{z^2+\dfrac{1}{z^2}}\)
b) \(B=\sqrt{x^2+\dfrac{1}{y^2}+\dfrac{1}{z^2}}+\sqrt{y^2+\dfrac{1}{z^2}+\dfrac{1}{x^2}}+\sqrt{z^2+\dfrac{1}{x^2}+\dfrac{1}{y^2}}\)
c) \(C=\sqrt{2x^2+\dfrac{3}{y^2}+\dfrac{4}{z}}+\sqrt{2y^2+\dfrac{3}{z^2}+\dfrac{4}{x^2}}+\sqrt{2z^2+\dfrac{3}{x^2}+\dfrac{4}{y^2}}\)
Cho x, y, z 0 thoả mãn x+y+z1. Chứng minh rằng:
a) sqrt{x^2+dfrac{1}{x^2}}+sqrt{y^2+dfrac{1}{y^2}}+sqrt{z^2+dfrac{1}{z^2}}gesqrt{82}
b) sqrt{x^2+dfrac{1}{x^2}+dfrac{1}{y^2}}+sqrt{y^2+dfrac{1}{y^2}+dfrac{1}{z^2}}+sqrt{z^2+dfrac{1}{z^2}+dfrac{1}{x^2}}gesqrt{163}
c)sqrt{x^2+dfrac{2}{y^2}+dfrac{3}{z^2}}+sqrt{y^2+dfrac{2}{z^2}+dfrac{3}{x^2}}+sqrt{z^2+dfrac{2}{z^2}+dfrac{3}{y^2}}gesqrt{406}
Đọc tiếp
Cho x, y, z > 0 thoả mãn x+y+z=1. Chứng minh rằng:
a) \(\sqrt{x^2+\dfrac{1}{x^2}}+\sqrt{y^2+\dfrac{1}{y^2}}+\sqrt{z^2+\dfrac{1}{z^2}}\ge\sqrt{82}\)
b) \(\sqrt{x^2+\dfrac{1}{x^2}+\dfrac{1}{y^2}}+\sqrt{y^2+\dfrac{1}{y^2}+\dfrac{1}{z^2}}+\sqrt{z^2+\dfrac{1}{z^2}+\dfrac{1}{x^2}}\ge\sqrt{163}\)
c)\(\sqrt{x^2+\dfrac{2}{y^2}+\dfrac{3}{z^2}}+\sqrt{y^2+\dfrac{2}{z^2}+\dfrac{3}{x^2}}+\sqrt{z^2+\dfrac{2}{z^2}+\dfrac{3}{y^2}}\ge\sqrt{406}\)
Đề bài: ax,y,z 0 và sqrt{x}+sqrt{y}+sqrt{z}1. Tìm Min P dfrac{x^3}{y+z}+dfrac{y^3}{z+x}+dfrac{z^3}{x+y}.
ĐÁP ÁN:
Ta có: dfrac{x^3}{y+z}+dfrac{y+z}{36}+dfrac{1}{162}+dfrac{y^3}{x+z}+dfrac{x+z}{36}+dfrac{1}{162}+dfrac{z^3}{x+y}+dfrac{x+y}{36}+dfrac{1}{162}ge3sqrt[3]{dfrac{x^3}{y+z}.dfrac{y+z}{36}.dfrac{1}{162}}+3sqrt[3]{dfrac{y^3}{x+z}.dfrac{x+z}{36}.dfrac{1}{162}}+3sqrt[3]{dfrac{z^3}{x+y}.dfrac{x+y}{36}.dfrac{1}{162}}3sqrt[3]{dfrac{x^3}{36.162}}+3sqrt[3]{dfrac{y^3}{36.162}}+3sqrt[3]{dfrac{z^3}{...
Đọc tiếp
Đề bài: ax,y,z >0 và \(\sqrt{x}+\sqrt{y}+\sqrt{z}=1\). Tìm Min P= \(\dfrac{x^3}{y+z}+\dfrac{y^3}{z+x}+\dfrac{z^3}{x+y}\).
ĐÁP ÁN:
Ta có: \(\dfrac{x^3}{y+z}+\dfrac{y+z}{36}+\dfrac{1}{162}+\dfrac{y^3}{x+z}+\dfrac{x+z}{36}+\dfrac{1}{162}+\dfrac{z^3}{x+y}+\dfrac{x+y}{36}+\dfrac{1}{162}\ge3\sqrt[3]{\dfrac{x^3}{y+z}.\dfrac{y+z}{36}.\dfrac{1}{162}}+3\sqrt[3]{\dfrac{y^3}{x+z}.\dfrac{x+z}{36}.\dfrac{1}{162}}+3\sqrt[3]{\dfrac{z^3}{x+y}.\dfrac{x+y}{36}.\dfrac{1}{162}}=3\sqrt[3]{\dfrac{x^3}{36.162}}+3\sqrt[3]{\dfrac{y^3}{36.162}}+3\sqrt[3]{\dfrac{z^3}{36.162}}=\dfrac{x+y+z}{6}.\)
=> P+\(\dfrac{x+y+z}{18}+\dfrac{1}{54}\)≥\(\dfrac{x+y+z}{6}\) <=> P≥\(\dfrac{x+y+z}{6}-\dfrac{x+y+z}{18}-\dfrac{1}{54}\)=\(\dfrac{x+y+z}{9}-\dfrac{1}{54}\)
Ta c/m đc: 3(x+y+z)≥(\(\sqrt{x}+\sqrt{y}+\sqrt{z}\))2 <=> 2(x+y+z) ≥2\(\left(\sqrt{xy}+\sqrt{xz}+\sqrt{yz}\right)\)<=> x+y+z≥\(\sqrt{xy}+\sqrt{xz}+\sqrt{yz}\)(luôn đúng)
➩x+y+z ≥ \(\dfrac{\left(\sqrt{x}+\sqrt{y}+\sqrt{z}\right)^3}{3}=\dfrac{1}{3}\) => P≥\(\dfrac{1}{54}\). Dấu ''='' xảy ra <=> x=y=z=\(\dfrac{1}{9}\)
Tìm bộ ba số thực x, y, z thỏa mãn: \(\dfrac{2}{\sqrt{x}+2\sqrt{y}+3\sqrt{z}}-\dfrac{1}{2\sqrt{xy}+6\sqrt{yz}+3\sqrt{xz}}=\dfrac{1}{3}\)
cho các số thực x,y,z thoả mãn x+y+z≥6.
Tìm minP=\(\dfrac{x^2}{yz+\sqrt{1+x^3}}+\dfrac{y^2}{xz+\sqrt{1+y^3}}+\dfrac{z^2}{xy+\sqrt{1+z^3}}\)
Cho mng tham khảo ạ
Cho x,y,z >0 thỏa mãn \(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=3\). Tìm GTLN của biểu thức \(P=\dfrac{1}{\sqrt{5x^2+2xy+2y^2}}+\dfrac{1}{\sqrt{5y^2+2yz+2z^2}}+\dfrac{1}{\sqrt{5z^2+2xz+2x^2}}\)
Cho x,y,z>0 /xyz=8.
Tìm min P= \(\dfrac{x^2}{\sqrt{\left(1+x^3\right)\left(1+y^3\right)}}+\dfrac{y^2}{\sqrt{\left(1+y^3\right)\left(1+z^3\right)}}+\dfrac{z^2}{\sqrt{\left(1+z^3\right)\left(1+x^3\right)}}\)
Giải phương trình: \(\sqrt{x-2}+\sqrt{y+2009}+\sqrt{z-2010}=\frac{1}{2}(x+y+z)\)