Violympic toán 9
Các câu hỏi tương tự
Cho a, b, c > 0. Chứng minh rằng: \(2\left(\frac{a}{b+2c}+\frac{b}{c+2a}+\frac{c}{a+2b}\right)\ge1+\frac{b}{b+2a}+\frac{c}{c+2b}+\frac{a}{a+2c}\)
Cho a,b,c >0 . Chứng minh rằng : \(\frac{a}{b+2c}+\frac{b}{c+2a}+\frac{c}{a+2b}+\frac{2a}{b+2a}+\frac{2b}{c+2b}+\frac{2c}{a+2c}\)≥3
18 tháng 11 2019 lúc 20:12
1.left{{}begin{matrix}a,b,c0ab+bc+ca3end{matrix}right. Cmr: frac{1}{a^2+1}+frac{1}{b^2+1}+frac{1}{c^2+1}gefrac{3}{2}
2.a,b,c0. Cmr: frac{ab^2}{a^2+2b^2+c^2}+frac{bc^2}{b^2+2c^2+a^2}+frac{ca^2}{c^2+2a^2+b^2}lefrac{a+b+c}{4}
3. a,b,c0. Cmr: frac{ab}{a+3b+2c}+frac{bc}{b+3c+2a}+frac{ca}{c+3a+2b}lefrac{a+b+c}{6}
Đọc tiếp
1.\(\left\{{}\begin{matrix}a,b,c>0\\ab+bc+ca=3\end{matrix}\right.\) Cmr: \(\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\ge\frac{3}{2}\)
2.\(a,b,c>0\). Cmr: \(\frac{ab^2}{a^2+2b^2+c^2}+\frac{bc^2}{b^2+2c^2+a^2}+\frac{ca^2}{c^2+2a^2+b^2}\le\frac{a+b+c}{4}\)
3. \(a,b,c>0\). Cmr: \(\frac{ab}{a+3b+2c}+\frac{bc}{b+3c+2a}+\frac{ca}{c+3a+2b}\le\frac{a+b+c}{6}\)
cho a, b, c > 0 và \(a^2+b^2+c^2=3\) C/m \(\frac{a}{a^2+2b+3}+\frac{b}{b^2+2c+3}+\frac{c}{c^2+2a+3}\le\frac{1}{2}\)
Cho a,b,c>0 và \(a^2b+b^2c+c^2a=3\)
Chứng minh rằng : \(\frac{ab+bc+ca}{2\left(a^2+b^2+c^2\right)}+\frac{1}{6}\left(\frac{a}{bc}+\frac{b}{ac}+\frac{c}{ab}\right)\)≥\(\frac{a+b+c}{3}\)
Cho a, b, c là các số thực dương thoả mãn \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\le2\). Chứng minh rằng: \(\frac{1}{\sqrt{5a^2+2ab+2b^2}}+\frac{1}{\sqrt{5b^2+2bc+2c^2}}+\frac{1}{\sqrt{5c^2+2ca+2a^2}}\le\frac{2}{3}\)
Cho a, b, c > 0. Chứng minh \(\frac{a}{2a+b+c}+\frac{b}{a+2b+c}+\frac{c}{a+b+2c}\le\frac{3}{4}\)
cho a,b,c>0; p=a+b+c Chứng minh \(\frac{1}{p-a}+\frac{1}{p-b}+\frac{1}{p-c}\ge2\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)
Cho a,b,c là số thực dương thỏa a+b+c=3 . Chứng minh \(\frac{1}{2 +a^2b}+\frac{1}{2+b^2c}+\frac{1}{2+c^2a}\ge1\)
Cho a,b,c > 0. CMR:
\(2\left(\frac{a}{b+2c}+\frac{b}{c+2a}+\frac{c}{a+2b}\right)\ge1+\frac{b}{b+2a}+\frac{c}{c+2b}+\frac{a}{a+2c}\)