áp dụng bất đẳng thức bu nhi a
ta có \(3\left(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\right)\ge\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^2\)
lại có a/b+b/c+c/a \(\ge\)3 (bđt cauchy)
nhân từng vế ta có \(3\left(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\right)\left(\frac{a}{b}+\frac{b}{a}+\frac{a}{c}\right)\ge3\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^2\)
suy ra đpcm
a/Cho a,b,c>0 . CMR: \(\frac{a}{1+a^2}+\frac{b}{1+b^2}+\frac{c}{1+c^2}\le\frac{3}{2}\)
b/ Cho\(a\ge b\ge c>0\)
CMR: \(\frac{a^2-b^2}{c}+\frac{c^2-b^2}{a}+\frac{a^2-c^2}{b}\ge3a-4b+c\)
CMR: nếu a>0, b>0, c>0 thì ta có:\(\frac{a^2}{b^2+c^2}+\frac{b^2}{c^2+a^2}+\frac{c^2}{a^2+b^2}\ge\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\)
Cho a,b,c >0. CMR:
\(\frac{a}{b^2}+\frac{b}{c^2}+\frac{c}{a^2}\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\)
Cho a+b>0; b+c>0, c+a>0. CMR:
\(\frac{a^3}{b+c}+\frac{b^3}{c+a}+\frac{c^3}{a+b}\ge\frac{a^2+b^2+c^2}{2}\)
Cho a,b,c >0. CMR
\(\frac{a^3+b^3+c^3}{2abc}+\frac{a^2+b^2}{ab+c^2}+\frac{b^2+c^2}{bc+a^2}+\frac{c^2+a^2}{ca+b^2}\ge\frac{9}{2}\)
Cho a;b;c > 0 thỏa mãn a + b + c = 1
CMR: \(\frac{ab}{a^2+b^2}+\frac{bc}{b^2+c^2}+\frac{ca}{c^2+a^2}+\frac{1}{4}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge\frac{15}{4}\)
cmr \(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\)
với mọi a,b,c khác 0
cm=cauchy
Cho a,b,c >0 a+b+c=3 CMR
\(\frac{a^2}{a+b^2}+\frac{b^2}{b+c^2}+\frac{c^2}{c+a^2}\ge\frac{3}{2}\)
Cho 3 số thực a , b , c. CMR :
\(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\)
vs a , b , c khác 0
