\(A=\frac{2016a}{ab+2016a+2016}+\frac{b}{bc+b+2016}+\frac{c}{ac+c+1}\)

\(A=\frac{2016a}{ab+2016a+abc}+\frac{b}{bc+b+2016}+\frac{bc}{abc+bc+b}\)

\(A=\frac{2016a}{a\left(b+2016+bc\right)}+\frac{b}{bc+b+2016}+\frac{bc}{2016+bc+b}\)

\(A=\frac{2016}{b+2016+bc}+\frac{b}{bc+b+2016}+\frac{bc}{2016+bc+b}\)

\(A=\frac{2016+b+bc}{2016+b+bc}=1\)

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 !

\(\frac{a}{ab+a+2016}+\frac{b}{bc+b+1}+\frac{2016c}{ac+2016c+2016}\)

\(=\frac{a}{ab+a+abc}+\frac{b}{bc+b+1}+\frac{abc^2}{ac+abc^2+abc}\)

\(=\frac{a}{a.\left(b+1+bc\right)}+\frac{b}{bc+b+1}+\frac{abc^2}{ac.\left(1+bc+b\right)}\)

\(=\frac{1}{b+bc+1}+\frac{b}{b+bc+1}+\frac{bc}{b+bc+1}\)

\(=\frac{1+b+bc}{b+bc+1}=1\)

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

\(\Rightarrow\frac{b+c+d}{a}=\frac{c+d+a}{b}=\frac{d+a+b}{c}=\frac{a+b+c}{d}\)

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

\(\Rightarrow\frac{a+b+c+d}{a}=\frac{a+b+c+d}{b}=\frac{a+b+c+d}{c}=\frac{a+b+c+d}{d}\)

\(\Rightarrow a=b=c=d\)Thao vào A ta được :

\(A=\frac{a^{2016}}{a^{2016}}+\frac{a^{2016}}{a^{2016}}+\frac{a^{2016}}{a^{2016}}+\frac{a^{2016}}{a^{2016}}=1+1+1+1=4\)

Ta co:

\(\text{ }P=\Sigma_{cyc}\frac{ab}{2016-c}=\Sigma_{cyc}\frac{ab}{a+b}\le\Sigma_{cyc}\frac{\frac{\left(a+b\right)^2}{4}}{a+b}=\Sigma_{cyc}\frac{a+b}{4}=1008\)

Dau '=' xay ra khi \(a=b=c=672\)

Tớ ko bt

\(\frac{a}{b}=\frac{c}{d}\Leftrightarrow\frac{d}{b}=\frac{c}{a}\Leftrightarrow\frac{d^{2016}}{b^{2016}}=\frac{c^{2016}}{a^{2016}}=\frac{c^{2016}-d^{2016}}{a^{2016}-b^{2016}}=\frac{c^{2016}+d^{2016}}{a^{2016}+b^{2016}}\)

(áp dụng tính chất dãy tỉ số bằng nhau)

Suy ra \(\frac{a^{2016}+b^{2016}}{a^{2016}-b^{2016}}.\frac{c^{2016}-d^{2016}}{c^{2016}+d^{2016}}=\frac{a^{2016}+b^{2016}}{c^{2016}+d^{2016}}.\frac{c^{2016}-d^{2016}}{a^{2016}-b^{2016}}\)

\(=\frac{b^{2016}}{d^{2016}}.\frac{d^{2016}}{b^{2016}}=1\)