Áp dụngk BĐt cô-si, ta có
\(\frac{a^2}{b^2c}+\frac{b^2}{c^2a}+\frac{1}{a}\ge3.\frac{1}{c}\)
Tương tự , rồi cộng vào, ta có
\(2A+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge3\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\Rightarrow A\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\left(ĐPCM\right)\)
^_^
Bài 1: CMR:
\(a,\dfrac{a}{b^2}+\dfrac{b}{c^2}+\dfrac{c}{a^2}\ge\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\)
\(b,\dfrac{a^3}{b\left(2c+a\right)}+\dfrac{b^3}{c\left(2a+b\right)}+\dfrac{c^3}{a\left(2b+c\right)}\ge1\) với a+b+c=3
Bài 2: \(a,b,c\in N,a+b+c=2021\)
Tìm GTNN \(P=\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\)
Cho \(\dfrac{a}{2a+b+c}+\dfrac{b}{2b+a+c}+\dfrac{c}{2c+a+b}=1\)
Tính \(\dfrac{a^2}{2a+b+c}+\dfrac{b^2}{2b+a+c}+\dfrac{c^2}{2c+a+b}\)
\(\dfrac{1}{2a-1}+\dfrac{1}{2b-1}+\dfrac{1}{2c-1}+3\ge\dfrac{4}{a+b}+\dfrac{4}{b+c}+\dfrac{4}{c+a}\)
Với a, b, c \(\ge\) 1
Cho các số thực dương a,b,c thỏa mãn a+b+c=3. CMR: \(\dfrac{1}{2+a^2b}+\dfrac{1}{2+b^2c}+\dfrac{1}{2+c^2a}\) ≥ 1
Cho a, b,c : abc = 1. Chứng minh:
\(\dfrac{a^2b^2}{2a^2+b^2+3a^2b^2}+\dfrac{b^2c^2}{2b^2+c^2+3b^2c^2}+\dfrac{c^2a^2}{2c^2+a^2+3a^2c^2}\le\dfrac{1}{2}\)
a, a,b,c>0. CMR:\(\dfrac{ab}{a+b+2c}+\dfrac{bc}{b+c+2a}+\dfrac{ac}{a+c+2b}\le\dfrac{a+b+c}{4}\)
b, a,b,c>0. CMR:\(\dfrac{ab}{a+3b+2c}+\dfrac{bc}{b+3c+2a}+\dfrac{ac}{c+3a+2b}\le\dfrac{a+b+c}{6}\)
Bài 5: cho a,b,c lớn hơn 0
chứng minh rẳng:
\(2\left(\dfrac{a}{b+2c}+\dfrac{b}{c+2a}+\dfrac{c}{a+2b}\right)\ge1+\dfrac{b}{b+2a}+\dfrac{c}{c+2b}+\dfrac{a}{a+2c}\)
cho a,b,c là các số thực dương thỏa mãn \(a+b+c+1=4abc\).CMR
\(\dfrac{a^2b}{b+2c}+\dfrac{b^2c}{c+2a}+\dfrac{c^2a}{a+2b}\ge1\)
cho a,b,c>0 thỏa mãn \(a^2+b^2+c^2=1\).CMR
\(\dfrac{\sqrt{ab+2c^2}}{\sqrt{1+ab-c^2}}+\dfrac{\sqrt{bc+2a^2}}{\sqrt{1+bc-a^2}}+\dfrac{\sqrt{ca+2b^2}}{\sqrt{1+ca-b^2}}\ge2+ab+bc+ca\)
