Cho a,b,c > 0 và abc = 1
CMR: \(\frac{2}{a^2\left(b+c\right)}+\frac{2}{b^2\left(a+c\right)}+\frac{2}{c^2\left(a+b\right)}\ge3\)
Cho \(a\ge1;b\ge2;c\ge3\)
CMR: \(\left(a-1\right)^2+\left(b-2\right)^2+\left(c-3\right)^2\le3\left(b-2\right)\)
Tìm Min của: \(P=\frac{1}{a^2}+\frac{4}{b^2}+\frac{8}{c^2}\)
Cho a,b,c,d>0 thỏa abcd=1. CMR \(\frac{a^3}{b^2\left(c^2+d^2\right)}+\frac{b^3}{c^2\left(d^2+a^2\right)}+\frac{c^3}{d^2\left(a^2+b^2\right)}+\frac{d^3}{a^2\left(b^2+c^2\right)}\ge2\)
Cho a,b,c là các số thực dương và abc = 1
CMR: \(\left(a+\frac{1}{b}\right)^2+\left(b+\frac{1}{c}\right)^2+\left(c+\frac{1}{a}\right)^2\ge3\left(a+b+c+1\right)\)
CMR: \(\left(2+\frac{a}{b}\right)^{\alpha}+\left(2+\frac{b}{c}\right)^{\alpha}+\left(2+\frac{c}{a}\right)^{\alpha}\ge3^{\alpha+1}\left(\forall a,b,c>0\right)\)
Bài 1:Cho a,b,c,d là các số dương. Chứng minh rằng :
\(\frac{a^4}{\left(a+b\right)\left(a^2+b^2\right)}+\frac{b^4}{\left(b+c\right)\left(b^2+c^2\right)}+\frac{c^4}{\left(c+d\right)\left(c^2+d^2\right)}+\frac{d^4}{\left(d+a\right)\left(d^2+a^2\right)}\ge\frac{a+b+c+d}{4}\)
Bài 2:Cho \(a>0,b>0,c>0\).\(CM:\frac{a}{bc}+\frac{b}{ca}+\frac{c}{ab}\ge2\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)
Bài 3: a) Cho x,y,>0. CMR:\(\frac{x^3}{x^2+xy+y^2}\ge\frac{2x-y}{3}\)
b) Chứng minh rằng\(\Sigma\frac{a^3}{a^2+ab+b^2}\ge\frac{a+b+c}{3}\)
Cho các số dương a, b, c thỏa mãn ab+bc+ca=1.
CMR: \(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\ge3+\sqrt{\frac{\left(a+b\right)\left(a+c\right)}{a^2}}+\sqrt{\frac{\left(b+c\right)\left(b+a\right)}{b^2}}+\sqrt{\frac{\left(c+a\right)\left(c+b\right)}{c^2}}\)
1) Cho a, b, c > 0. CMR: \(a^2+b^2+c^2+abc+5\ge3\left(a+b+c\right)\)
2) Cho a, b, c > 0, đặt \(x=a+\frac{1}{b}\), \(y=b+\frac{1}{c}\), \(z=c+\frac{1}{a}\). Chứng minh rằng: \(xy+yz+zx\ge2\left(x+y+z\right)\)
3) Cho các số dương x, y, z thỏa mãn xyz = 1. Chứng minh rằng: \(x^2+y^2+z^2+x+y+z\ge2\left(xy+yz+zx\right)\)
haiz...
Cho a,b,c>0 và a+b+c=3.CMR
\(\frac{\left(1+a\right)^2\left(1+b\right)^2}{1+c^2}+\frac{\left(1+b\right)^2\left(1+c\right)^2}{1+a^2}+\frac{\left(1+c\right)^2\left(1+a\right)^2}{1+b^2}\ge24\)