Đặt \(A=\left(\dfrac{c}{a}+\dfrac{c}{b}\right)^2\)
\(A=\left[c\left(\dfrac{1}{a}+\dfrac{1}{b}\right)\right]^2\)
\(A=c^2\cdot\left(\dfrac{1}{a}+\dfrac{1}{b}\right)^2\)
\(A\ge\left(a^2+b^2\right)\cdot\dfrac{16}{\left(a+b\right)^2}\)
Cần cm:\(a^2+b^2\ge\dfrac{\left(a+b\right)^2}{2}\)
\(\Leftrightarrow\left(a-b\right)^2\ge0\)(luôn đúng)
\(\Rightarrowđpcm\)
Cho a, b, c là ba số dương thoả mãn abc = 1. Chứng minh rằng: \(\dfrac{1}{a^3\left(b+c\right)}+\dfrac{1}{b^3\left(c+a\right)}+\dfrac{1}{c^3\left(a+b\right)}\ge\dfrac{3}{2}\)
Cho 3 số a, b, c thỏa mãn a # -b, b # -c, c # -a.
Chứng minh rằng : \(\dfrac{a^2-bc}{\left(a+b\right)\left(a+c\right)}+\dfrac{b^2-ac}{\left(a+b\right)\left(b+c\right)}+\dfrac{c^2-ab}{\left(c+a\right)\left(c+b\right)}=0\)
Cho a,b,c dương. Chứng minh :
\(\dfrac{a^2+b^2}{a+b}+\dfrac{b^2+c^2}{b+c}+\dfrac{c^2+a^2}{c+a}\le\dfrac{3\left(a^2+b^2+c^2\right)}{a+b+c}\)
Cho 3 số thực a, b, c đôi một khác nhau thỏa mãn: \(\dfrac{a}{b-c}+\dfrac{b}{c-a}+\dfrac{c}{a-b}=0\)
CMR: \(\dfrac{a}{\left(b-c\right)^2}+\dfrac{b}{\left(c-a\right)^2}+\dfrac{c}{\left(a-b\right)^2}=0\)
Cho a, b, c là các số dương thỏa mãn \(\left(a^2+1\right)\left(b^2+4\right)\left(c^2+9\right)=48abc\)
Tính \(\dfrac{a^3+b^3+c^3}{\left(a+b+c\right)^2}\)
Cho \(\dfrac{a-\left(c-b\right)}{b-c}+\dfrac{b-\left(a-c\right)}{c-a}+\dfrac{c-\left(b-a\right)}{a-b}=3\)
Chứng minh rằng: \(\dfrac{a}{\left(b-c\right)^2}+\dfrac{b}{\left(c-a\right)^2}+\dfrac{c}{\left(a-b\right)^2}=0\)
chứng minh
\(2\left(\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\right)-3=\dfrac{\left(a-b\right)^2}{\left(a+c\right)\left(b+c\right)}+\dfrac{\left(b-c\right)^2}{\left(b+a\right)\left(c+a\right)}+\dfrac{\left(c-a\right)^2}{\left(c+b\right)\left(a+b\right)}\)
cho các số dương a và b thỏa mãn a+b=1. chứng minh rằng \(\left(1+\dfrac{1}{a}\right)\left(1+\dfrac{1}{b}\right)\ge9\)
Chứng minh
\(\dfrac{\left(a-b\right)^2}{ab}+\dfrac{\left(b-c\right)^2}{bc}+\dfrac{\left(c-a\right)^2}{ca}=\left(a+b+c\right)\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)\)
