Cho a,b,c khác 0 và đôi 1 khác nhau t/m a+b+c=0. Tính
A=\(\left(\dfrac{a-b}{c}+\dfrac{b-c}{a}+\dfrac{c-a}{b}\right)\left(\dfrac{c}{a-b}+\dfrac{a}{b-c}+\dfrac{b}{c-a}\right)\)
Cho a, b, c > 0. Chứng minh: \(\left(a+\dfrac{1}{b}-1\right)\left(b+\dfrac{1}{c}-1\right)+\left(b+\dfrac{1}{c}-1\right)\left(c+\dfrac{1}{a}-1\right)+\left(c+\dfrac{1}{a}-1\right)\left(a+\dfrac{1}{b}-1\right)\ge3\)
1. Biết a, b, c đôi 1 khác nhau . Chứng miinh rằng :
\(\dfrac{b-c}{\left(a-b\right)\left(a-c\right)}+\dfrac{c-a}{\left(b-c\right)\left(b-a\right)}+\dfrac{a-b}{\left(c-a\right)\left(c-b\right)}=\dfrac{2}{a-b}+\dfrac{2}{b-c}+\dfrac{2}{c-a}\).
2. Cho x,y,z đôi một khác nhau thoả mãn \(\dfrac{xy+1}{y}+\dfrac{yz+1}{z}+\dfrac{zx+1}{x}\). Chứng minh rằng : \(\left|xyz\right|=1\)
cho các số thực không âm a , b , c ( a khác b ) thỏa mãn (a+c)(b+c)=1
Tìm min A \(\dfrac{1}{\left(a-b\right)^2}\)+\(\dfrac{1}{\left(a+c\right)^2}\)+\(\dfrac{1}{\left(b+c\right)^2}\)
cho a,b,c là các số thực dương. Chứng minh rằng :
\(\dfrac{b^2c}{a^3\left(b+c\right)}+\dfrac{c^2a}{b^3\left(c+a\right)}+\dfrac{a^2b}{c^3\left(a+b\right)}\ge\dfrac{1}{2}\left(a+b+c\right)\)
Cho a, b, c là độ dài 3 cạnh tam giác. CMR:
1, \(\dfrac{1}{\left(a+b-c\right)^n}+\dfrac{1}{\left(a-b+c\right)^n}+\dfrac{1}{\left(b+c-a\right)^n}\ge\dfrac{1}{a^n}+\dfrac{1}{b^n}+\dfrac{1}{c^n}\)
2, \(\dfrac{1}{a^n}+\dfrac{1}{b^n}+\dfrac{1}{c^n}\ge4^n\left[\dfrac{1}{\left(2a+b+c\right)^n}+\dfrac{1}{\left(a+2b+c\right)^n}+\dfrac{1}{\left(a+b+2c\right)^n}\right]\)
Cho \(a,b,c\ge1\). CMR:
\(a\left(b+c\right)+b\left(c+a\right)+c\left(a+b\right)+2\left(\dfrac{1}{a^2+1}+\dfrac{1}{b^2+1}+\dfrac{1}{c^2+1}\right)\ge9\)
Cho a,b,c là các số nguyên dương. CM:
a) \(\left(a,b,c\right)=\dfrac{\left(a,b,c\right)abc}{\left(a,b\right)\left(b,c\right)\left(c,a\right)}\)
b) \(\left[a,b,c\right]=\dfrac{\left(a,b,c\right)\left[a,b\right]\left[b,c\right]\left[c,a\right]}{abc}\)
Cho a,b,c \(\ge1.CMR:a\left(b+c\right)+b\left(c+a\right)+c\left(a+b\right)+2\left(\dfrac{1}{1+a^2}+\dfrac{1}{1+b^2}+\dfrac{1}{1+c^2}\right)\ge9\)
