Do  a < b < c < d < m < n 
=> 2c < c + d 
m< n => 2m < m+ n 
=> 2c + 2a +2m = 2 ( a + c + m) < a +b + c + d + m + n) 
Do đó :
(a + c + m)/(a + b + c + d + m + n) < 1/2(đcpcm)

Do  a < b < c < d < m < n 
=> 2c < c + d 
m< n => 2m < m+ n 
=> 2c + 2a +2m = 2 ( a + c + m) < a +b + c + d + m + n) 
Do đó :
\(\dfrac{\text{(a + c + m)}}{\left(a+b+c+d+m+n\right)}\) < \(\dfrac{1}{2}\)

Ta có :

a < b \(\Rightarrow\)2a < a + b \(\Rightarrow\)\(\frac{a}{a+b}< \frac{1}{2}\)

c < d \(\Rightarrow\)2c < c + d \(\Rightarrow\)\(\frac{c}{c+d}< \frac{1}{2}\)

m < n \(\Rightarrow\)2m < m + n \(\Rightarrow\)\(\frac{m}{m+n}< \frac{1}{2}\)

\(\Rightarrow\)2a + 2c + 2m < ( a + b ) + ( c + d ) + ( m + n ) 

\(\Rightarrow\)2 . (a  + c + nm ) < a + b + c + d + m + n

\(\Rightarrow\)\(\frac{a+c+m}{a+b+c+d+m+n}< \frac{1}{2}\)

\(a< b\Rightarrow2a< a+b\)

\(c< d\Rightarrow2c< c+d\)

\(m< n\Rightarrow2m< m+n\)

\(\Rightarrow2a+2c+2m< a+b+c+d+m+n\)

\(\Rightarrow2\left(a+c+m\right)< a+b+c+d+m+n\)

\(\Rightarrow\frac{a+c+m}{a+b+c+d+m+n}< \frac{1}{2}\left(\text{đ}pcm\right)\)

Ta thấy:
\(\hept{\begin{cases}a< b\\c< d\\m< n\end{cases}\Rightarrow a+c+m< b+d+n}\)
\(\Rightarrow\left(a+c+m\right)+\left(a+c+m\right)< \left(a+c+m\right)+\left(b+d+n\right)\)
\(\Rightarrow2\left(a+c+m\right)< a+b+c+d+m+n\)
hay \(a+b+c+d+m+n>2\left(a+c+m\right)\)
\(\Rightarrow\frac{a+c+m}{a+b+c+d+m+n}< \frac{a+c+m}{2\left(a+c+m\right)}\) ( do các tử và các mẫu đều dương )
\(\Rightarrow\frac{a+c+m}{a+b+c+d+m+n}< \frac{1}{2}\) ( đpcm )

Sửa  đề c/m \(\frac{a+c+m}{a+b+c+d+m+n}< \frac{1}{2}\)

Ta có: \(\hept{\begin{cases}a< b\Rightarrow2a< a+b\\c< d\Rightarrow2c< c+d\\m< n\Rightarrow2m< m+n\end{cases}}\)

=>\(2\left(a+c+m\right)< a+b+c+d+m+n\)

=>\(\frac{a+c+m}{a+b+c+d+m+n}< \frac{1}{2}\)

Sửa đề: Chứng minh: \(\frac{a+c+m}{a+b+c+d+m+n}< \frac{1}{2}\)

Ta có: \(\hept{\begin{cases}a< b\Rightarrow2a< a+b\\c< d\Rightarrow2c< c+d\\m< n\Rightarrow2m< m+n\end{cases}}\)

\(\Rightarrow2\left(a+c+m\right)< a+b+c+d+m+n\)

\(\Rightarrow\frac{a+c+m}{a+b+c+d+m+n}< \frac{1}{2}\)

Đề sai cho mình sửa lại :

Cho 6 số nguyên dương a < b < c < d < m < n

Chứng minh rằng \(\frac{a+c+m}{a+b+c+d+m+n}<\frac{1}{2}\)

Bài giải:

Ta có :a < b \(\Rightarrow\) 2a < a + b   ;  c < d \(\Rightarrow\) 2c < c + d  ;  m < n \(\Rightarrow\) 2m < m + n

Suy ra 2a + 2c + 2m = 2(a + c + m) < (a + b + c + d + m + n). Do đó

Vậy : \(\frac{a+c+m}{a+b+c+d+m+n}<\frac{1}{2}\)  (đpcm)

do a<b<c<d<m<n

=> a+c+m < b+d+n

=> 2(a+c+m) < a+b+c+d+m+n

=> \(\frac{2\left(a+c+m\right)}{a+b+c+d+m+n}< 1\)  => \(\frac{a+c+m}{a+b+c+d+m+n}< \frac{1}{2}\)

Câu hỏi của Đinh Trần Nhật Minh - Toán lớp 7 - Học toán với OnlineMath

Em tham khảo nhé!

ta có 

a<b<c=>3a<a+b+c

d<m<n=>3d<d+m+n

=>3a+3d<a+b+c+d+m+n

=>3a+3a/a+b+c+d+m+n<a+b+c+m+n+d/a+b+c+d+m+n

=>3(a+d)/a+b+c+d+m+n)<1

=>a+d/a+b+c+d+m+n<1/3  (đpcm)

copy

a<b<c<d<m<n =>a+b+c+d+m+n>a+b+a+b+a+b=3(a+b)

\(\Rightarrow\frac{a+b}{a+b+c+d+m+n}

do a<b<c<d<m<n

=>a+b<c+d

a+b<m+n

=>a+b+a+b+a+b<a+b+c+d+m+n

=>a+b+a+b+a+b/a+b+c+d+m+n<a+b+c+d+m+n/a+b+c+d+m+n

<=>3(a+b)/a+b+c+m+d+n<1

=>a+b/a+b+c+d+m+b<1/3  (đpcm)

a<b<c<d<m<n thì:

a+b+c > 3a ; d+m+n > 3d => a+b+c+d+m+n > 3a + 3d

Do đó: \(\frac{a+d}{a+b+c+d+m+n}< \frac{a+d}{3a+3d}=\frac{1}{3}.\)đpcm