Violympic toán 9
Các câu hỏi tương tự
Cho a,b,c \(\ge\)0 TM \(a^2+b^2+c^2=1\) CM
\(\frac{c}{1+ab}+\frac{b}{1+ac}+\frac{a}{1+bc}\ge1\)
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 và a+b+c ≤ 1. CMR: A = \(\frac{1}{a^2+2bc}+\frac{1}{b^2+2ca}+\frac{1}{c^2+2ab}\) ≥ 9
Cho a,b,c 0 . Cmr :
a, a+b+frac{1}{4}≥sqrt{a+b}
b, ( a+b+ frac{1}{4})2 + ( b+c+frac{1}{4}) + ( c+a+frac{1}{4}) ≥ (frac{1}{frac{1}{a}+frac{1}{b}}+frac{1}{frac{1}{b}+frac{1}{c}}+frac{1}{frac{1}{c}+frac{1}{a}})
Đọc tiếp
Cho a,b,c > 0 . Cmr :
a, a+b+\(\frac{1}{4}\)≥\(\sqrt{a+b}\)
b, ( a+b+ \(\frac{1}{4}\))2 + ( b+c+\(\frac{1}{4}\)) + ( c+a+\(\frac{1}{4}\)) ≥ (\(\frac{1}{\frac{1}{a}+\frac{1}{b}}\)+\(\frac{1}{\frac{1}{b}+\frac{1}{c}}\)+\(\frac{1}{\frac{1}{c}+\frac{1}{a}}\))
a) cho x,y dương. CMR: \(\frac{1}{x}+\frac{1}{y}\ge\frac{4}{x+y}\)
b) cho a+b+c=1 CMR: \(\frac{a}{a+b^2}+\frac{b}{b+c^2}+\frac{c}{c+a^2}\le\frac{1}{4}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)
cho các số thực dương a, b,c t/m
\(a+b+c\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.\) Chứng minh
\(a+b+c\ge\frac{3}{a+b+c}+\frac{2}{abc}\)
Cho a , b , c là ba cạnh của tam giác. CMR:
\(\frac{a}{b+c-a}+\frac{b}{a+c-b}+\frac{c}{a+b-c}\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\)
28 tháng 7 2019 lúc 18:33
Cho: a,b,c > 0 và a + b + c = 3.
Chứng minh rằng:
a) \(\frac{a+b}{1+a}+\frac{b+c}{1+b}+\frac{c+a}{1+c}\ge ab+bc+ca\)
b) \(\frac{a}{ab+b^3}+\frac{b}{bc+c^3}+\frac{c}{ca+a^3}\ge\frac{3}{2}\)
cho a,b>0 cm\(\frac{1}{1+a^2}+\frac{1}{1+b^2}\ge\frac{2}{1+ab}\) nếu \(ab\ge1\)
b) cho a,b,c\(\ge\)1. CMR \(\frac{1}{1+a^4}+\frac{1}{1+b^4}+\frac{1}{1+c^4}\ge\frac{1}{1+ab^3}+\frac{1}{1+bc^3}+\frac{1}{1+ca^3}\)