Nhìn cái đề đã thấy người ra đề vui tính. \(a+b+c=2009\)

1 trong a;b;c là 2009 nghĩa là 2 số bằng 0

\(\dfrac{1}{a};\dfrac{1}{b};\dfrac{1}{c}\) hoán vị của \(\dfrac{1}{0};\dfrac{1}{0};\dfrac{1}{2009}\)

và \(\dfrac{1}{0}=?\)

Bài 1: 

Ta có: \(a+b\ge2\sqrt{ab}\)

\(b+c\ge2\sqrt{bc}\)

\(a+c\ge2\sqrt{ac}\)

Do đó: \(2\left(a+b+c\right)\ge2\left(\sqrt{ab}+\sqrt{bc}+\sqrt{ac}\right)\)

hay \(a+b+c\ge\sqrt{ab}+\sqrt{cb}+\sqrt{ac}\)

ĐKXĐ : a;b;c  \(\ne0\)

Ta có : \(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=\dfrac{1}{2000}\)

\(\Leftrightarrow\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=\dfrac{1}{a+b+c}\)

\(\Leftrightarrow\dfrac{1}{b}+\dfrac{1}{c}=\dfrac{1}{a+b+c}-\dfrac{1}{a}\)

\(\Leftrightarrow\dfrac{b+c}{bc}=\dfrac{-\left(b+c\right)}{a\left(a+b+c\right)}\)

\(\Leftrightarrow\left(b+c\right)\left(\dfrac{1}{bc}+\dfrac{1}{a\left(a+b+c\right)}\right)=0\)

\(\Leftrightarrow\left(b+c\right).\dfrac{a\left(a+b+c\right)+bc}{abc\left(a+b+c\right)}=0\)

\(\Leftrightarrow\left(b+c\right).\dfrac{a^2+ab+ac+bc}{abc\left(a+b+c\right)}=0\)

\(\Leftrightarrow\dfrac{\left(b+c\right)\left(a+b\right)\left(a+c\right)}{abc\left(a+b+c\right)}=0\)

\(\Leftrightarrow\left[{}\begin{matrix}b+c=0\\a+b=0\\a+c=0\end{matrix}\right.\left(1\right)\)

Từ (1) kết hợp a + b + c = 2000 ta được điều phải chứng minh

\(B=\sqrt{1+2008^2+\dfrac{2008^2}{2009^2}}+\dfrac{2008}{2009}=\sqrt{\dfrac{2009^2+2008^2.2009^2+2008^2}{2009^2}}+\dfrac{2008}{2009}=\dfrac{\sqrt{2009^2+\left(2009-1\right)^2.2009^2+2008^2}}{2009}+\dfrac{2008}{2009}=\dfrac{\sqrt{2009^2+2009^4-2.2009.2009^2+2009^2+2008^2}+2008}{2009}=\dfrac{\sqrt{2009^4+2.2009^2-2.\left(2008+1\right).2009^2+2008^2}+2008}{2009}=\dfrac{\sqrt{2009^4+2.2009^2-2.2008.2009^2-2.2009^2+2008^2}+2008}{2009}=\dfrac{\sqrt{2009^4-2.2008.2009^2+2008^2}+2008}{2009}=\dfrac{\sqrt{\left(2009^2-2008\right)^2}+2008}{2009}=\dfrac{2009^2-2008+2008}{2009}=2009\in N\)

Vậy B có giá trị là một số tự nhiên

Xét các số thực a, b, c thỏa mãn \(a+b+c=0\)

\(\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}=\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{2}{ab}+\frac{2}{bc}+\frac{2}{ca}-\frac{2}{ab}-\frac{2}{bc}-\frac{2}{ca}}\)

\(=\sqrt{\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2-2.\frac{a+b+c}{abc}}\)

\(=\sqrt{\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2}=\left|\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right|\)

Ta có:

\(B=\sqrt{1+2008^2+\frac{2008^2}{2009^2}}+\frac{2008}{2009}\)

\(=\sqrt{2008^2}.\sqrt{\frac{1}{2018^2}+\frac{1}{1^2}+\frac{1}{2009^2}}+\frac{2008}{2009}\)

\(=2008.\sqrt{\frac{1}{2018^2}+\frac{1}{1^2}+\frac{1}{\left(-2009\right)^2}}+\frac{2008}{2009}\)

\(=2008.\left|\frac{1}{2008}+1-\frac{1}{2009}\right|+\frac{2008}{2009}\)

\(=2008.\left(\frac{1}{2008}+1-\frac{1}{2009}\right)+\frac{2008}{2009}\)

\(=2008.\left(\frac{1}{2008}+1-\frac{1}{2009}+\frac{1}{2009}\right)\)

\(=2008.\frac{2009}{2008}=2009\in\text{N}\)

\(A=\sqrt{1+2008^2+\dfrac{2008^2}{2009^2}}+\dfrac{2008}{2009}=\sqrt{2008^2+2.2008+1-2.2008+\dfrac{2008^2}{2009^2}}+\dfrac{2008}{2009}=\sqrt{2009^2-2.2009.\dfrac{2008}{2009}+\dfrac{2008^2}{2009^2}}+\dfrac{2008}{2009}=\sqrt{\left(2009-\dfrac{2008}{2009}\right)^2}+\dfrac{2008}{2009}=2009\)

Vậy , A có giá trị là số nguyên .

Đặt \(2008=a\)

\(\Leftrightarrow A=\sqrt{1+a^2+\dfrac{a^2}{\left(a+1\right)^2}}+\dfrac{a}{a+1}\\ A=\sqrt{\left(a+1\right)^2-\dfrac{2a\left(a+1\right)}{a+1}+\dfrac{a^2}{\left(a+1\right)^2}}+\dfrac{a}{a+1}\\ A=\sqrt{\left(a+1-\dfrac{a}{a+1}\right)^2}+\dfrac{a}{a+1}\\ A=a+1-\dfrac{a}{a+1}+\dfrac{a}{a+1}=a+1=2009\left(đpcm\right)\)

`A=\sqrt{1+2008^2+2008^2/2009^2}+2008/2009`

`=\sqrt{1+2008^2+2.2008+2008^2/2009^2-2.2008}+2008/2009`

`=\sqrt{(2008+1)^2-2.2008+2008^2/2009^2}+2008/2009`

`=\sqrt{2009-2.2008/2009*2009+2008^2/2009^2}+2008/2009`

`=\sqrt{(2009-2008/2009)^2}+2008/2009`

`=|2009-2008/2009|+2008/2009`

`=2009-2008/2009+2008/2009`

`=2009` là 1 số tự nhiên