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\(A=\frac{2016a}{ab+2016a+2016}+\frac{b}{bc+b+2016}+\frac{c}{ac+c+1}\)
\(A=\frac{2016a}{ab+2016a+abc}+\frac{b}{bc+b+2016}+\frac{bc}{abc+bc+b}\)
\(A=\frac{2016a}{a\left(b+2016+bc\right)}+\frac{b}{bc+b+2016}+\frac{bc}{2016+bc+b}\)
\(A=\frac{2016}{b+2016+bc}+\frac{b}{bc+b+2016}+\frac{bc}{2016+bc+b}\)
\(A=\frac{2016+b+bc}{2016+b+bc}=1\)
Thay : 2016 = abc
ta có :
\(A=\frac{a^2bc}{ab+a^2bc+abc}+\frac{b}{bc+b+abc}+\frac{c}{ac+c+1}\)
\(A=\frac{a^2bc}{ab\left(1+ac+c\right)}+\frac{b}{b\left(c+1+ac\right)}+\frac{c}{ac+c+1}\)
\(A=\frac{ac}{ac+c+1}+\frac{1}{ac+c+1}+\frac{c}{ac+c+1}\)
\(A=\frac{ac+c+1}{ac+c+1}\)
\(A=1\)
vậy \(A=\frac{2016.a}{ab+2016.a+2016}+\frac{b}{bc+b+2016}+\frac{c}{ac+c+1}=1\)
Chúc bạn học tốt !
Làm đơn giản thế này thôi nhé An Kì :
Ta có : \(2016a+bc=\left(a+b+c\right)a+bc=a^2+ab+ac+bc=a\left(a+b\right)+c\left(a+b\right)=\left(a+b\right)\left(a+c\right)\)Tương tự : \(2016b+ac=\left(a+b\right)\left(b+c\right)\)
\(2016c+ab=\left(a+c\right)\left(b+c\right)\)
\(\Rightarrow\left(2016a+bc\right)\left(2016b+ac\right)\left(2016c+ab\right)=\left(a+b\right)^2\left(b+c\right)^2\left(c+a\right)^2\)
Ta có : \(2016a+bc=\left(a+b+c\right).a+bc=a^2+ab+ac+bc=a\left(a+b\right)+c\left(a+b\right)=\left(a+b\right)\left(a+c\right)\)
\(2016b+ac=\left(a+b+c\right).b+ac=ab+b^2+bc+ac=b\left(a+b\right)+c\left(a+b\right)=\left(a+b\right)\left(b+c\right)\)
\(2016c+ab=\left(a+b+c\right)c+ab=ac+bc+c^2+ab=a\left(b+c\right)+c\left(b+c\right)=\left(a+c\right)\left(b+c\right)\)
\(\Rightarrow\left(2016a+bc\right)\left(2016b+ac\right)\left(2016c+ab\right)=\left(a+b\right)^2\left(b+c\right)^2\left(c+a\right)^2\) (đpcm)
a2 + b2 + c2 = ab + bc + ca
=>2.(a2+b2+c2)=2.(ab+bc+ca)
<=>a2+2b2+2c2=2ab+2bc+2ca
<=>2a2+2b2+2c2-2ab-2bc-2ca=0
<=>a2-2ab+b2+b2-2bc+c2+c2-2ca+a2=0
<=>(a-b)2+(b-c)2+(c-a)2=0
<=>a-b=0 và b-c=0 và c-a=0
<=>a=b và b=c và c=a
=> a=b=c
mà a;b;c khác 0 nên
P=1+1+1=3
a2 + b2 + c2 = ab + bc + ca => 2. (a2 + b2 + c2 )= 2.( ab + bc + ca)
<=> (a2 - 2ab + b2) + (b2 - 2bc + c2) + (c2 - 2ca + a2) = 0
<=> (a - b)2 + (b - c)2 + (c - a)2 = 0 <=> (a - b)2 = (b - c)2 = (c - a)2 = 0 (Vì (a - b)2 \(\ge\) 0; ( b - c)2 \(\ge\)0 ; (c - a)2 \(\ge\) 0
<=> a = b = c
=> \(P=\frac{a^4}{a^4}+\frac{b^4}{b^4}+\frac{a^{2016}}{a^{2016}}=1+1+1=3\)
\(\frac{a}{ab+a+2016}+\frac{b}{bc+b+1}+\frac{2016c}{ac+2016c+2016}\)
\(=\frac{a}{ab+a+abc}+\frac{b}{bc+b+1}+\frac{abc^2}{ac+abc^2+abc}\)
\(=\frac{a}{a.\left(b+1+bc\right)}+\frac{b}{bc+b+1}+\frac{abc^2}{ac.\left(1+bc+b\right)}\)
\(=\frac{1}{b+bc+1}+\frac{b}{b+bc+1}+\frac{bc}{b+bc+1}\)
\(=\frac{1+b+bc}{b+bc+1}=1\)