Ta thấy  ;1/19+1/29+1/31+1/39>1/40+1/40+1/40+1/40

              1/19+1/29+1/31+1/39>1/40.4=4/40=1/10

             Vậy 1/19+1/29+1/31+1/39>1/10

vì 1/19>1/40; 1/29>1/40; 1/31>1/40; 1/39>1/40

nên 1/19+1/29+1/31+1/39>1/40+1/40+1/40+1/40=1/10

nên M>1/10. Vậy M>1/10

M>\(\frac{1}{10}\)

ta có:

\(\frac{1}{19}>\frac{1}{40};\frac{1}{29}>\frac{1}{40};\frac{1}{31}>\frac{1}{40};\frac{1}{39}>\frac{1}{40}\)

\(\Rightarrow\frac{1}{19}+\frac{1}{29}+\frac{1}{31}+\frac{1}{39}>\frac{1}{40}.4\)=\(\frac{1}{10}\)

M = 1 /19 + 1 /29 + 1/ 31 + 1/ 39 > 1 / 40 + 1/ 40 + 1/ 40 + 1/40 = 4 / 40 = 1/10 nen 

M > 1/ 10

vì 1/9 > 1/40 ; 1/29 > 1/40 ; 1/31 > 1/40; 1/39 > 1/40

nên 1/9 + 1/ 29 + 1/31 + 1/39 > 1/40 + 1/40 + 1/40 + 1/40 mà 1/40 + 1/40 + 1/40 + 1/40 = 1/10 

=) M > 1/10

M > 1/20 + 1/30 + 1/40 + 1/40 

M> 2/15 > 2/20 = 1/10
=> M > 1/10

Nguyễn Thu Hiền 1/19 bạn ơi

\(M=\frac{10^{30}+1}{10^{31}+1}\)

\(\Rightarrow10M=\frac{10\cdot(10^{30}+1)}{10^{31}+1}\)

\(\Rightarrow10M=\frac{10^{31}+10}{10^{31}+1}\)

\(\Rightarrow10M=\frac{10^{31}+1+9}{10^{31}+1}\)

\(\Rightarrow10M=1+\frac{9}{10^{31}+1}\)

\(N=\frac{10^{31}+1}{10^{32}+1}\)

\(\Rightarrow10N=\frac{10\cdot(10^{31}+1)}{10^{32}+1}\)

\(\Rightarrow10N=\frac{10^{32}+10}{10^{32}+1}\)

\(\Rightarrow10N=\frac{10^{32}+1+9}{10^{32}+1}\)

\(\Rightarrow10N=1+\frac{9}{10^{32}+1}\)

Mà\(1+\frac{9}{10^{31}+1}>1+\frac{9}{10^{32}+1}\)

Nên \(10M>10N\)

Hay \(M>N\)