Ta có: \(\left(x+y\right)^2\ge4xy\)
\(\Rightarrow\frac{xy}{x+y}\le\frac{1}{4}\left(x+y\right)\)
\(\Rightarrow\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\le\frac{1}{4}\left(a+b\right)+\frac{1}{4}\left(b+c\right)+\frac{1}{4}\left(c+a\right)\)
\(=\frac{a+b+c}{2}\)
Dấu \("="\) xảy ra \(\Leftrightarrow a=b=c\)
Cho a,b,c>0, abc\(=\)1
CMR: \(\frac{1}{a^2-ab+b^2}+\frac{1}{b^2-bc+c^2}+\frac{1}{c^2-ca+a^2}\le a+b+c\)
Cho a,b,c là các số thực thỏa mãn \(a^2+b^2+c^2=1.CMR:\frac{bc}{a^2+1}+\frac{ca}{b^2+1}+\frac{ab}{c^2+1}\le\frac{3}{4}\)
Cho a, b, c > 0. CMR:
a) \(\frac{a^2}{a+b}\) + \(\frac{b^2}{b+c}\) + \(\frac{c^2}{c+a}\) ≥ \(\frac{ab}{a+b}\) + \(\frac{bc}{b+c}\) + \(\frac{ca}{c+a}\)
b) a2b2(a2 + b2) ≤ 2 với a + b = 2 và a; b > 0
Cho a, b, c là các số dương. Chứng minh rằng:
1. Cho a,b,c > 0. Cmr :
\(\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}\ge\frac{3\left(a^2+b^2+c^2\right)}{a+b+c}\)
2. Cho a,b,c > 0. Cmr :
\(\frac{a}{b+2c+3d}+\frac{b}{c+2d+3a}+\frac{c}{d+2a+3b}+\frac{d}{a+2b+3c}\ge\frac{2}{3}\)
Cho a,b,c thỏa mãn \(\frac{1}{a^2+2}+\frac{1}{b^2+2}+\frac{1}{c^2+2}\) = 1
a, Tính \(\frac{a^2}{a^2+2}+\frac{b^2}{b^2+2}+\frac{c^2}{c^2+2}\)
b, CMR ab + bc +ca ≤ 3
Cho a, b, c > 0 và \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1\)
CMR : \(\frac{a^2}{a+bc}+\frac{b^2}{b+ca}+\frac{c^2}{c+ab}\) ≥ \(\frac{a+b+c}{4}\)
Cho a,b,c là các số thực dương thỏa mãn \(a^2+b^2+c^2=1\). CMR
\(\frac{bc}{a^2+1}+\frac{ca}{b^2+1}+\frac{ab}{c^2+1}\le\frac{3}{4}\)
Cho a,b,c>0 thỏa mãn ab+bc+ca=2020
Cmr:\(\frac{a-b}{2020+c^2}+\frac{b-c}{2020+a^2}+\frac{c-a}{2020+b^2}\)
