Cho a,b,c thực dương .CMR
\(\sqrt{\frac{\left(a+b\right)^3}{ab\left(4a+4b+c\right)}}+\sqrt{\frac{\left(b+c\right)^3}{bc\left(4b+4c+a\right)}}+\sqrt{\frac{\left(c+a\right)^3}{ca\left(4c+4c+b\right)}}\ge2\sqrt{2}\)
Cho a,b,c là các số thực dương. CMR:
\(\frac{1}{2a+b+\sqrt{8bc}}-\frac{8}{\sqrt{2b^2+2\left(a+c\right)^2+3}}\ge\frac{-3}{2}\)
CHo a,b,c > 0 thỏa mãn: abc=1 .CMR:
\(\frac{1}{a^3\left(b+c\right)}+\frac{1}{b^3\left(a+c\right)}+\frac{1}{c^3\left(a+b\right)}\ge\frac{3}{2}\) (1)
Cho a , b , c là ba số thực dương. Chứng minh bất đẳng thức:
\(\frac{1}{a\left(a^2+8bc\right)}+\frac{1}{b\left(b^2+2ca\right)}+\frac{1}{c\left(c^2+2ab\right)}\le\frac{1}{3abc}\)
Chứng minh rằng :
a) \(\frac{\left(a+b\right)^2}{2}+\frac{a+b}{4}\ge a\sqrt{b}+b\sqrt{a}\)với \(a,b\ge0\)
b) \(\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}>2\)với \(a,b,c>0\)
Cho a,b,c>0 thỏa mãn a+b+c = 3. CMR:
\(25\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)+351\ge88\left(a^2+b^2+c^2\right)\)
Cho các số thực dương a, b, c thỏa mãn \(a^2+b^2+c^2+abc=4\).Chứng minh rằng: \(\frac{1}{2}< \frac{a}{4-bc}+\frac{b}{4-ca}+\frac{c}{4-ab}\le1\)
\(A=\left(6:\frac{3}{5}-1\frac{1}{6}x\frac{6}{7}\right):\left(4\frac{1}{5}x\frac{10}{11}+5\frac{2}{11}\right)\)\(B=\left(1-\frac{1}{2}\right)x\left(1-\frac{1}{4}\right)x.......x\left(1-\frac{1}{2015}\right)x\left(1-\frac{1}{2016}\right)\)
\(C=5\frac{9}{10}:\frac{3}{2}-\left(2\frac{1}{3}x4\frac{1}{2}-2x2\frac{1}{3}\right):\frac{7}{4}\)
Tính:
a,\(\frac{\left(3+\frac{1}{6}\right)-\frac{2}{5}}{\left(5-\frac{1}{6}\right)+\frac{7}{10}}\)
b,\(\frac{\left(4,08-\frac{2}{25}\right):\frac{4}{17}}{\left(6\frac{5}{9}-3\frac{1}{4}\right)x2\frac{2}{7}}\)
c,\(\frac{2-\frac{1}{4}+\frac{1}{3}-\frac{3}{5}}{3-\frac{1}{5}-\frac{5}{3}}\)