[tex]\frac{ab}{a^{2}+ab+bc}+\frac{bc}{b^{2}+bc+ca}+\frac{ca}{c^{2}+ca+ab}\leq 1[/tex]
[tex]A=\frac{ab}{a^{2}+ab+bc}+\frac{bc}{b^{2}+bc+ca}+\frac{ca}{c^{2}+ca+ab}\\\\ =\frac{1}{\frac{a^2+ab+bc}{ab}}+\frac{1}{\frac{b^2+bc+ca}{bc}}+\frac{1}{\frac{c^2+ca+ab}{ca}}\\\\ =\frac{1}{\frac{a}{b}+1+\frac{c}{a}}+\frac{1}{\frac{b}{c}+1+\frac{a}{b}}+\frac{1}{\frac{c}{a}+1+\frac{b}{c}}\\\\ (\frac{a}{b};\frac{b}{c};\frac{c}{a})=(x;y;z) =>xyz=1\\\\ => A=\frac{1}{x+1+z}+\frac{1}{y+1+a}+\frac{1}{z+1+b}\\\\ (\sqrt[3]{x};\sqrt[3]{y};\sqrt[3]{z})=(m;n;p) =>mnp=1\\\\ => A=\frac{1}{m^3+1+p^3}+\frac{1}{n^3+1+m^3}+\frac{1}{p^3+1+n^3}\\\\ \leq \frac{1}{mp.(m+p)+1}+\frac{1}{mn.(m+n)+1}+\frac{1}{np.(n+p)+1}\\\\ => A\leq \frac{n}{m+n+p}+\frac{p}{m+n+p}+\frac{m}{m+n+p}=1[/tex]
dấu "=" <=> a=b=c
[TẶNG BẠN] TRỌN BỘ Bí kíp học tốt 08 môn
