giai ho minh voi

Ta có :  \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2017}\)

\(\Leftrightarrow\frac{bc+ac+ab}{abc}=\frac{1}{a+b+c}\)( do a + b + c = 2017 )

\(\Rightarrow\left(a+b+c\right)\left(bc+ac+ab\right)=abc\)

\(\Leftrightarrow\left(bc+ac\right)\left(a+b+c\right)+ab\left(a+b\right)+abc-abc=0\)

\(\Leftrightarrow c\left(a+b\right)\left(a+b+c\right)+ab\left(a+b\right)=0\)

\(\Leftrightarrow\left(a+b\right)\left(ac+bc+c^2+ab\right)=0\)

\(\Leftrightarrow\left(a+b\right)\left[b\left(c+a\right)+c\left(c+a\right)\right]=0\)

\(\Leftrightarrow\left(a+b\right)\left(b+c\right)\left(c+a\right)=0\)

Ta có : hoặc a+b =0

            hoặc b+c =0

           hoặc c+a = 0 

Mà  \(a+b+c=2017\)

\(\Rightarrow\)hoặc a = 2017; hoặc b = 2017 ; hoặc c = 2017

Vậy ...

\(a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\)=> \(a+b+c=\frac{ab+bc+ac}{abc}=ab+bc+ac\)

Ta có \(\left(a-1\right)\left(b-1\right)\left(c-1\right)=\left(abc-1\right)+a+b+c-ab-bc-ac=0\)

=> có ít nhất 1 trong 3 số a,b,c bằng 1

Vậy có ít nhất 1 trong 3 số a,b,c bằng 1

biến đổi tương đương đưa về (a-1)(b-1)(c-1)=0

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

\(\Leftrightarrow a+b+c=\frac{ab+bc+ac}{abc}\)

\(\Leftrightarrow a+b+c=ab+bc+ac\left(abc=1\right)\)

\(\Leftrightarrow1+a+b+c-ab-bc-ac-1=0\)

\(\Leftrightarrow abc+a+b+c-ab-bc-ac-1=0\)

\(\Leftrightarrow ab\left(c-1\right)-a\left(c-1\right)-b\left(c-1\right)+c-1=0\)

\(\Leftrightarrow\left(ab-a-b+1\right)\left(c-1\right)=0\)

\(\Leftrightarrow\left(a-1\right)\left(b-1\right)\left(c-1\right)=0\)

\(\Leftrightarrow\)a = 1 hoặc b = 1 hoặc c = 1

=> Đpcm 

Thay 1 = abc ta có: \(a+b+c=\frac{abc}{a}+\frac{abc}{b}+\frac{abc}{c}\)

<=> a + b + c = bc + ac + ab

<=> (a - ac) + (b - bc) + (c - ab) = 0 

<=> a(1 - c) + b(1 - c) + (c - \(\frac{1}{c}\)) = 0 

<=> ca(1 - c) + cb(1 - c) + (c - 1)(c + 1) = 0 

<=> (1 - c)(ca + cb - c - 1) = 0 

<=> (1 - c)[c(a -1) + (cb - abc)]= 0 

<=> (1 - c)[c(a - 1) + cb(1 - a)]= 0 

<=> (1 - c)(a - 1)(c - cb) = 0

<=> (1 - c)(a - 1)(1 - b).c = 0 <=> a = 1 hoặc b = 1 hoặc c = 1

Vậy.... 

http://olm.vn/hoi-dap/question/179947.html

Cso bài mk đó,lik-e nha

Đặt \(\left(\frac{a}{b^2},\frac{b}{c^2},\frac{c}{a^2}\right)=\left(x,y,z\right)\)

\(\Rightarrow xyz=\frac{abc}{a^2b^2c^2}=\frac{1}{abc}=1\)

Theo bài ra ta có : \(\frac{a}{b^2}+\frac{b}{c^2}+\frac{c}{a^2}=\frac{a^2}{c}+\frac{b^2}{a}+\frac{c^2}{b}\)

\(\Leftrightarrow x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\)

\(\Leftrightarrow x+y+z=xy+yz+xz\)

\(\Leftrightarrow\left(xy-x-y+1\right)-1+z\left(x+y-1\right)=0\)

\(\Leftrightarrow\left(xy-x-y+1\right)+z\left(x+y-1-xy\right)=0\)

\(\Leftrightarrow\left(x-1\right)\left(y-1\right)-z\left(x-1\right)\left(y-1\right)=0\)

\(\Leftrightarrow\left(x-1\right)\left(y-1\right)\left(1-z\right)=0\)

\(\Leftrightarrow\frac{a-b^2}{b^2}.\frac{b-c^2}{c^2}.\frac{a^2-c}{a^2}=0\)

\(\Leftrightarrow\left(a-b^2\right)\left(b-c^2\right)\left(c-a^2\right)=0\)

Ta có đpcm

Thay a+b+c=2017 vào \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2017}\)  ta có:

\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}\)

\(\Rightarrow\frac{1}{a}+\frac{1}{b}+\frac{1}{c}-\frac{1}{a+b+c}=0\)

\(\Rightarrow\frac{a+b}{ab}+\frac{a+b+c-c}{c\left(a+b+c\right)}=0\)\(\Rightarrow\frac{a+b}{ab}+\frac{a+b}{c\left(a+b+c\right)}=0\)

\(\Rightarrow\left(a+b\right)\left(\frac{1}{ab}+\frac{1}{c\left(a+b+c\right)}\right)=0\)\(\Rightarrow\left(a+b\right)\left(\frac{c\left(a+b+c\right)+ab}{abc\left(a+b+c\right)}\right)=0\)

\(\Rightarrow\left(a+b\right)\left(\frac{c\left(b+c\right)+ca+ab}{abc\left(a+b+c\right)}\right)=0\)

\(\Rightarrow\left(a+b\right)\left[c\left(b+c\right)+ca+ab\right]=0\)

\(\Rightarrow\left(a+b\right)\left[c\left(b+c\right)+a\left(b+c\right)\right]=0\)

\(\Rightarrow\left(a+b\right)\left(b+c\right)\left(c+a\right)=0\)

\(\Rightarrow\)\(a+b=0\) hoặc \(b+c=0\) hoặc \(c+a=0\)

\(\Rightarrow\)\(c=2017\)hoặc \(a=2017\) hoặc \(b=2017\left(đpcm\right)\)

Ta có 1/a + 1/b + 1/c = (bc + ac + ac)/abc = ab + bc + ca 
=> a + b + c = ab + bc + ca 
<=> a + b + c - ab - bc - ca = 0 
<=> a + b + c - ab - bc - ac + abc - 1 = 0 
<=> (a - ab) + (b - 1) + (c - bc) + (abc - ac) = 0 
<=> -a(b - 1) + (b - 1) - c(b - 1) + ac(b - 1) = 0 
<=> (b - 1)(-a + 1 -c + ac) = 0 
<=> (b - 1)[ (-a + 1) + (ac - c) ] = 0 
<=> (b - 1)[ -(a - 1) + c(a - 1) ] = 0 
<=> (a - 1)(b - 1)(c - 1) = 0 
<=> a - 1 = 0 hoặc b - 1 = 0 hoặc c - 1 = 0 
<=> a = 1 hoặc b = 1 hoặc c = 1 
 

Từ abc=1=>c=1/ab

Và a+b+c=1/a+1/b+1/c

<=>a+b+1/ab=1/a+1/b+ab

<=>ab-a-b+1-(1/ab-1/a-1/b+1)=0

<=>a(b-1)-(b-1)-1/a(1/b-1)-(1/b-1)=0

<=>(b-1)(a-1)-(1/b-1)(1/a-1)=0

<=>(a-1)(b-1)-(1-b/b)(1-a/a)=0

<=>(a-1)(b-1)-(a-1)(b-1)/ab=0

<=>(a-1)(b-1)(1-1/ab)=0

<=>(a-1)(b-1)(c-1)=0

<=>a-1=0 hoặc b-1=0 hoặc c-1=0

=>a=1 hoặc b=1 hoặc c=1 (đpcm)

Với đk a, b,c khác 0

a+b+c=1<=> a+b=1-c

Ta có \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1\Leftrightarrow bc+ac+ba=abc\Leftrightarrow c\left(b+a\right)+ab\left(1-c\right)=0\)

<=> \(c\left(1-c\right)+ab\left(1-c\right)=0\Leftrightarrow\left(1-c\right)\left(c+ab\right)=0\Leftrightarrow\left(1-c\right)\left(1-a-b+ab\right)=0\)

<=>\(\left(1-c\right)\left[\left(1-a\right)-b\left(1-a\right)\right]=0\Leftrightarrow\left(1-a\right)\left(1-b\right)\left(1-c\right)=0\Leftrightarrow\)a=1 hoặc b=1 hoặc c=1