CM \(\frac{mbc+n}{\left(a-b\right)\left(a-c\right)}+\frac{mac+n}{\left(b-a\right)\left(b-c\right)}+\frac{mab+n}{\left(c-a\right)\left(c-b\right)}=m\)\(m\)
(n bất kỳ, còn a, b, c đôi 1 \(\ne\)nhau)
Cho a , b , c , n là các số dương
CMR \(a^{\left(n+1\right)\left(b+c\right)}+b^{\left(n+1\right)\left(a+c\right)}+c^{\left(n+1\right)\left(a+b\right)}\ge\frac{a^n+b^n+c^n}{2}\)
Cho a,b,c là các số thực không âm và n ≥ log23 - 1. Chứng minh rằng :
\(\left(\frac{a}{b+c}\right)^n+\left(\frac{b}{c+a}\right)^n+\left(\frac{c}{a+b}\right)^n+\frac{\left(2^{n+1}-3\right)abc}{2^{n-3}\left(a+b\right)\left(b+c\right)\left(c+a\right)}\ge2\)
Cho a,b,c là 3 số thực đôi một phân biệt. CMR:
\(3+\frac{\left(2a+b\right)\left(2b+c\right)}{\left(a-b\right)\left(b-c\right)}+\frac{\left(2b+c\right)\left(2c+a\right)}{\left(b-c\right)\left(c-a\right)}+\frac{\left(2c+a\right)\left(2a+b\right)}{\left(c-a\right)\left(a-b\right)}=\frac{2a+b}{a-b}+\frac{2b+c}{b-c}+\frac{2c+a}{c-a}\)
cho a,b,c khác nhau CMR:
\(\frac{b-c}{\left(a-b\right)\left(a-c\right)}+\frac{c-a}{\left(b-c\right)\left(b-a\right)}+\frac{a-b}{\left(c-a\right)\left(c-b\right)}=\frac{2}{b-c}+\frac{2}{c-a}+\frac{2}{a-b}.\)
\(n\ge3;n\inℕ\)
CMR:
\(\frac{1}{a^n\left(b+c\right)}+\frac{1}{b^n\left(c+a\right)}+\frac{1}{c^n\left(a+b\right)}\ge\frac{3}{2}\)
Cho ab + bc + ca = 3abc.
CMR \(\frac{\left(a+b\right)\left(a+c\right)+\left(b+c\right)\left(b+a\right)+\left(c+a\right)\left(c+b\right)}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}=\frac{3}{2}\)
Cho a,b,c>0 và a+b+c=3
CMR: \(\frac{a^3}{\left(a+b\right)\left(a+c\right)}+\frac{b^3}{\left(b+c\right)\left(b+a\right)}+\frac{c^3}{\left(c+a\right)\left(c+b\right)}\ge\frac{3}{4}\)
CMR: Nếu a,b,c khác nhau thì:
\(\frac{b-c}{\left(a-b\right)\left(a-c\right)}+\frac{c-b}{\left(b-c\right)\left(b-a\right)}+\frac{a-b}{\left(c-a\right)\left(c-b\right)}=\frac{2}{a-b}+\frac{2}{b-c}+\frac{2}{c-a}\)