Violympic toán 9
Các câu hỏi tương tự
chứng minh rằng với mọi số dương a,b,c ta luôn có
\(\frac{1}{a\left(1+b\right)}+\frac{1}{b\left(1+c\right)}+\frac{1}{c\left(1+a\right)}\ge\frac{3}{1+abc}\)
chứng minh rằng với mọi số dương a,b,c ta luôn có
\(\frac{1}{a\left(1+b\right)}+\frac{1}{b\left(1+c\right)}+\frac{1}{c\left(1+a\right)}\ge\frac{3}{1+abc}\)
Cho a,b,c dương và abc=1
CMR: \(\frac{a^4}{2\left(b+c\right)^2}+\frac{b^4}{2\left(a+c\right)^2}+\frac{c^4}{2\left(a+b\right)^2}+\frac{1}{c^2\left(a+c\right)\left(a+b\right)}+\frac{1}{b^2\left(a+b\right)\left(b+c\right)}+\frac{1}{a^2\left(a+c\right)\left(a+b\right)}\ge\frac{1}{8}\)
cho a, b, c là các số thực dương thảo mãn abc=1 chứng minh rằng \(\frac{a}{\left(a+1\right)\left(b+1\right)}+\frac{b}{\left(a+1\right)\left(c+1\right)}+\frac{c}{\left(b+1\right)\left(a+1\right)}\ge\frac{3}{4}\)
1. chứng minh với mọi a, b, c dương ta luôn có \(\frac{1}{a\left(1+b\right)}+\frac{1}{b\left(1+c\right)}+\frac{1}{c\left(1+a\right)}\ge\frac{3}{1+abc}\)
2. tìm x nguyên để \(x^4-x^3+2x+2\) là số chính phương
Cho a, b, c dương thỏa abc = 1. Chứng minh: \(\frac{1}{a^3\left(7b+3c\right)}+\frac{1}{b^3\left(7c+3a\right)}+\frac{1}{c^3\left(7a+3b\right)}\ge\frac{1}{10}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)
Cho a,b,c 0. CMR:
1. a^3+b^3+c^3ge3abc
2. frac{x^2}{a}+frac{y^2}{b}gefrac{left(x+yright)^2}{a+b}
3. frac{x^2}{a}+frac{y^2}{b}+frac{z^2}{c}gefrac{left(x+y+zright)^2}{a+b+c}
4. a^4+b^4+c^4ge abcleft(a+b+cright)
5. frac{1}{left(a+1right)^2}+frac{1}{left(b+1right)^2}gefrac{1}{ab+1}
6.frac{1}{1+a^3}+frac{1}{1+b^3}+frac{1}{1+c^3}gefrac{3}{1+abc}
Đọc tiếp
Cho a,b,c > 0. CMR:
1. \(a^3+b^3+c^3\ge3abc\)
2. \(\frac{x^2}{a}+\frac{y^2}{b}\ge\frac{\left(x+y\right)^2}{a+b}\)
3. \(\frac{x^2}{a}+\frac{y^2}{b}+\frac{z^2}{c}\ge\frac{\left(x+y+z\right)^2}{a+b+c}\)
4. \(a^4+b^4+c^4\ge abc\left(a+b+c\right)\)
5. \(\frac{1}{\left(a+1\right)^2}+\frac{1}{\left(b+1\right)^2}\ge\frac{1}{ab+1}\)
6.\(\frac{1}{1+a^3}+\frac{1}{1+b^3}+\frac{1}{1+c^3}\ge\frac{3}{1+abc}\)
Cho a,b,c>0 thỏa mãn abc=1. Chứng minh
\(\frac{a}{\left(a+1\right)\left(b+1\right)}+\frac{b}{\left(b+1\right)\left(c+1\right)}+\frac{c}{\left(c+1\right)\left(a+1\right)}\ge\frac{3}{4}\)
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