Question

cho 3 số dương 0≤a≤b≤c≤1 chứng minh rằng (a/bc+1)+(b/bc+1)+(c/ab+1)≤2

Answer 1

Ta có: \(0\le a\le b\le1\Rightarrow\hept{\begin{cases}a-1\ge0\\b-1\ge0\end{cases}}\)

\(\Rightarrow\left(a-1\right)\left(b-1\right)\ge0\Leftrightarrow ab-a-b+1\ge0\)

\(\Leftrightarrow ab+1\ge a+b\Leftrightarrow\frac{c}{ab+1}\le\frac{c}{a+b}\)(Vì \(c\ge0\))

Mà \(\frac{c}{a+b}\le\frac{c+c}{a+b+c}=\frac{2c}{a+b+c}\)(Vì \(c\ge0\))

\(\Rightarrow\frac{c}{ab+1}\le\frac{2c}{a+b+c}\)

Chứng minh tương tự: \(\frac{b}{bc+1}\le\frac{2b}{a+b+c};\frac{c}{ab+1}\le\frac{2c}{a+b+c}\)

\(\Rightarrow\frac{a}{bc+1}+\frac{b}{bc+1}+\frac{c}{ab+1}\le\frac{2\left(a+b+c\right)}{a+b+c}=2\left(đpcm\right)\)