Cho a, b, c là độ dài 3 cạnh của 1 tam giác. CMR:
a) \(a^2+b^2+c^2< 2\left(ab+bc+ca\right)\)
b) \(\dfrac{a}{b+c-a}+\dfrac{b}{a+c-b}+\dfrac{c}{a+b-c}\ge3\)
c) \(\dfrac{1}{a+b},\dfrac{1}{b+c},\dfrac{1}{c+a}\)cũng là độ dài ba cạnh của 1 tam giác
Chứng minh
\(\dfrac{\left(a-b\right)^2}{ab}+\dfrac{\left(b-c\right)^2}{bc}+\dfrac{\left(c-a\right)^2}{ca}=\left(a+b+c\right)\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)\)
Cho a,b,c là ba số khác nhau và a+b+c=0. Cmr:\(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}=\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)^2\)
Cho a,b,c >0. Chứng minh rằng
\(\dfrac{1}{a^2+bc}+\dfrac{1}{b^2+ac}+\dfrac{1}{c^2+ab}\le\dfrac{a+b+c}{2abc}\)
Cho a, b, c > 0 .CMR: \(\dfrac{ab}{a+b}+\dfrac{bc}{b+c}+\dfrac{ac}{a+c}\) ≤ \(\dfrac{1}{2}\)
Cho a, b, c > 0 .CMR: \(\dfrac{ab}{a+b}+\dfrac{bc}{b+c}+\dfrac{ac}{a+c}\) ≤ \(\dfrac{1}{2}\left(a+b+c\right)\)
cho a, b, c \(\ne\) 0. CMR:
\(\dfrac{a^3-b^3}{ab^2}+\dfrac{b^3-c^3}{bc^2}+\dfrac{c^3-a^3}{ca^2}\ge0\)
1. Cho a,b > 0. CMR:
a) \(\dfrac{a}{b}+\dfrac{b}{a}+\dfrac{ab}{a^2-ab+b^2}\ge3\)
b) \(\dfrac{a}{b}+\dfrac{b}{a}+\dfrac{9ab}{a^2+b^2}\ge\dfrac{13}{2}\)
Các bạn ơi giúp mk với.
Cho a,b,c khác 0 thỏa mãn \(\left(1+\dfrac{b}{a}\right)\left(1+\dfrac{c}{b}\right)\left(1+\dfrac{a}{c}\right)=8\)
CMR \(\dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{c+a}=\dfrac{3}{4}+\dfrac{ab}{\left(a+b\right)\left(b+c\right)}+\dfrac{bc}{\left(b+c\right)\left(c+a\right)}+\dfrac{ca}{\left(c+a\right)\left(a+b\right)}\)