Question

cho a,b,c >o c/m
1<\(\frac{a}{a+b}+\frac{b}{b+c}+\frac{a}{c+a}\)<2

Answer 1

Sửa đề: \(1< \frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}< 2\)

\(\frac{a}{a+b}>\frac{a}{a+b+c};\frac{b}{b+c}>\frac{b}{a+b+c};\frac{c}{c+a}>\frac{c}{a+b+c}\)

Cộng vế với vế:

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

Lại có: \(\frac{a}{a+b}< \frac{a+c}{a+b+c}\) ; \(\frac{b}{b+c}< \frac{a+b}{a+b+c}\) ; \(\frac{c}{c+a}< \frac{b+c}{a+b+c}\)

Cộng vế với vế: \(\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}< \frac{2\left(a+b+c\right)}{a+b+c}=2\)