Question

CMR  \(M=\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\)ko the la so nguyen (với a, b, c nguyên)

Answer 1

Ta thấy:  \(\frac{a}{a+b}>\frac{a}{a+b+c}\)

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

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

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

=>\(M>1\) (1)

Lại có:

Áp dụng: Với \(\frac{m}{n}<1=>\frac{m}{n}<\frac{m+k}{n+k}\)(với \(m,n\in N\cdot\))

Ta có: \(\frac{a}{a+b}<1=>\frac{a}{a+b}<\frac{a+c}{a+b+c}\)

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

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

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

=>\(M<\frac{2.\left(a+b+c\right)}{a+b+c}\)

=>M<2 (2)

Từ (1) và (2)

=>1<M<2

=>M không phải số nguyên

=>ĐPCM