Question

cho a^2/b^2 +b^2/c^2 +c^2/a^2 = a/c +c/b +b/a cmr a=b=c

Answer 1

áp dụng bất đẳng thức côsi cho hai số dương

\(\frac{a^2}{b^2}+\frac{b^2}{c^2}\ge2\sqrt{\frac{a^2}{b^2}\cdot\frac{b^2}{c^2}}=2\frac{a}{c}\)

\(\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge2\frac{b}{a}\)

\(\frac{c^2}{a^2}+\frac{a^2}{b^2}\ge2\frac{c}{b}\)

cộng vế theo vế 

\(2\left(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\right)\ge2\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right)\)

dấu "=" xảy ra khi \(\frac{a^2}{b^2}=\frac{b^2}{c^2}=\frac{c^2}{a^2}\Leftrightarrow a=b=c\)