Violympic toán 9
Các câu hỏi tương tự
Cho a,b,c là 3 số thực khác 0 và thỏa mãn:
\(\left\{{}\begin{matrix}a^2\left(b+c\right)+b^2\left(c+a\right)+c^2\left(a+b\right)+2abc=0\\a^{2013}+b^{2013}+c^{2013}=1\end{matrix}\right.\)
Hãy tính giá trị của biểu thức: Q= \(\frac{1}{a^{2013}}+\frac{1}{b^{2013}}+\frac{1}{c^{2013}}\)
1. Tính gt của bt:
\(A=\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+...+\frac{1}{\sqrt{24}+\sqrt{25}}\)
2. Tính tổng \(S=\sqrt{1+\left(1+\frac{1}{3}\right)^2}+\sqrt{1+\left(\frac{1}{2}+\frac{1}{4}\right)^2}+\sqrt{1+\left(\frac{1}{3}+\frac{1}{5}\right)^2}+...+\sqrt{1+\left(\frac{1}{2014}+\frac{1}{2016}\right)^2}\)
BL
23 tháng 2 2020 lúc 11:23
1. a) Tìm nin N*, n2008 sao cho 2^{2008}+2^{2012}+2^{2013}+2^{2014}+2^{2016}+2^n là số chính phương
b) tìm x,y 0 thỏa mãn x^2+y^22left(x+yright)left(sqrt{x}+sqrt{y}-2right)
2. a) left{{}begin{matrix}age0a+bge1end{matrix}right.. Min Afrac{8a^2+b}{4a}+b^2
b) left{{}begin{matrix}a,bge0left(a-bright)^2a+b+2end{matrix}right.. Cmr: left(1+frac{a^3}{left(b+1right)^3}right)left(1+frac{b^3}{left(b+1right)^3}right)le9
c) x,y0;left(x+sqrt{1+x^2}right)left(y+sqrt{1+y^2}right)2020. Min P x + y
d) x,y,...
Đọc tiếp
1. a) Tìm \(n\in N\)*, \(n>2008\) sao cho \(2^{2008}+2^{2012}+2^{2013}+2^{2014}+2^{2016}+2^n\) là số chính phương
b) tìm x,y > 0 thỏa mãn \(x^2+y^2=2\left(x+y\right)\left(\sqrt{x}+\sqrt{y}-2\right)\)
2. a) \(\left\{{}\begin{matrix}a\ge0\\a+b\ge1\end{matrix}\right.\). Min \(A=\frac{8a^2+b}{4a}+b^2\)
b) \(\left\{{}\begin{matrix}a,b\ge0\\\left(a-b\right)^2=a+b+2\end{matrix}\right.\). Cmr: \(\left(1+\frac{a^3}{\left(b+1\right)^3}\right)\left(1+\frac{b^3}{\left(b+1\right)^3}\right)\le9\)
c) \(x,y>0;\left(x+\sqrt{1+x^2}\right)\left(y+\sqrt{1+y^2}\right)=2020\). Min P = x + y
d) \(x,y,z>0;\sqrt{x^2+y^2}+\sqrt{y^2+z^2}+\sqrt{z^2+x^2}=6\). Min \(P=\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}\)
e) \(\left\{{}\begin{matrix}x,y,z>0\\x+y+z+4xyz=4\end{matrix}\right.\) Cmr: \(\left(1+xy+\frac{y}{z}\right)\left(1+yz+\frac{z}{x}\right)\left(1+zx+\frac{x}{y}\right)\ge27\)
f) \(\left\{{}\begin{matrix}x,y,z\ge1\\3x^2+4y^2+5z^2=52\end{matrix}\right.\). Min P = x + y + z
g) \(x,y>0\). Min \(P=\frac{2}{\sqrt{\left(2x+y\right)^3+1}-1}+\frac{2}{\sqrt{\left(x+2y\right)^3+1}-1}+\frac{\left(2x+y\right)\left(x+2y\right)}{4}-\frac{8}{3\left(x+y\right)}\)
Giải phương trình, hệ phương trình:
a) \(\frac{\sqrt{x-2013}-1}{x-2013}+\frac{\sqrt{y-2014}-1}{y-2014}+\frac{\sqrt{z-2015}-1}{z-2015}=\frac{3}{4}\)
b) \(\left\{{}\begin{matrix}x^3+1=2y\\y^3+1=2x\end{matrix}\right.\)
c)\(\sqrt{x^2-3x+2}+\sqrt{x-3}=\sqrt{x-2}+\sqrt{x^2+2x-3}\)
d)\(5x-2\sqrt{x}\left(2+y\right)+y^2+1=0\)
Cho \(x=\sqrt{\dfrac{1}{2\sqrt{3}-2}-\dfrac{3}{2.\left(\sqrt{3}+1\right)}}\). Tính: \(A=\dfrac{4.\left(x+1\right).x^{2013}-2.x^{2012}+2x+1}{2x^2+3x}\)
Cho a,b,c>0 thỏa mãn: a.b.c=8
Chứng minh: \(\frac{a^2}{\sqrt{\left(1+a^3\right).\left(1+b^3\right)}}+\frac{b^2}{\sqrt{\left(1+b^3\right).\left(1+c^3\right)}}+\frac{c^2}{\sqrt{\left(1+c^3\right).\left(1+a^3\right)}}\ge\frac{4}{3}\)
BL
11 tháng 2 2020 lúc 21:04
1. left{{}begin{matrix}a,b,c0a+b+c1end{matrix}right.. Cmr: frac{ab}{sqrt{left(1-cright)^2left(1+cright)}}+frac{bc}{sqrt{left(1-aright)^2left(1+aright)}}+frac{ca}{sqrt{left(1-bright)^3left(1+bright)}}lefrac{3sqrt{2}}{8}
2. left{{}begin{matrix}a,b,c0a+b+cle1end{matrix}right.. Cmr: frac{1}{a^2+b^2+c^2}+frac{1}{ableft(a+bright)}+frac{1}{bcleft(b+cright)}+frac{1}{acleft(a+cright)}gefrac{87}{2}
3. left{{}begin{matrix}a,b,c0ab+bc+ca2abcend{matrix}right.. Cmr: frac{1}{aleft(2a-1right)^2}+frac{1}{bleft...
Đọc tiếp
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
1, gpt
a,sqrt{x^2-3x+2}+sqrt{x+3}sqrt{x-2}+sqrt{x^2+2x-3}
b, left(4x+2right)sqrt{x+8}3x^2+7x+8
c,sqrt{x+sqrt{2x-1}}+sqrt{x-sqrt{2x-1}}sqrt{2}
2/ cho x,y,z thỏa mãn : left(frac{1}{x}+frac{1}{y}+frac{1}{z}right):frac{1}{x+y+z}1
tính giá trị biểu thức Bleft(x^{29}+y^{29}right)left(x^{11}+y^{11}right)left(x^{2013}+y^{2013}right)
Đọc tiếp
1, gpt
a,\(\sqrt{x^2-3x+2}+\sqrt{x+3}=\sqrt{x-2}+\sqrt{x^2+2x-3}\)
b, \(\left(4x+2\right)\sqrt{x+8}=3x^2+7x+8\)
c,\(\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=\sqrt{2}\)
2/ cho x,y,z thỏa mãn : \(\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right):\frac{1}{x+y+z}=1\)
tính giá trị biểu thức B=\(\left(x^{29}+y^{29}\right)\left(x^{11}+y^{11}\right)\left(x^{2013}+y^{2013}\right)\)
1.CM đẳng thức: \(\sqrt[3]{\sqrt[3]{2}-1}=\sqrt[3]{\frac{1}{9}}+\sqrt[3]{\frac{2}{9}}+\sqrt[3]{\frac{4}{9}}\)
2. Giai hệ phương trình:
\(\left\{{}\begin{matrix}x^2\left(2013y-2012\right)=1\\x\left(y^2+2012\right)=2013\end{matrix}\right.\)