Question

Giúp tớ với pleaseeeeeee:

Cho \(0\le a,b,c,d\le1\)

CMR: \(\frac{a}{bcd+1}+\frac{b}{acd+1}+\frac{c}{abd+1}+\frac{d}{abc+1}\le3\)

Thanks mn

Answer 1

Vì \(0\le a,b,c,d\le1\Rightarrow abc+1\ge abcd+1\)

\(\Rightarrow VT\le\frac{a+b+c+c}{abcd+1}\)

Do \(\hept{\begin{cases}\left(1-a\right)\left(1-b\right)\ge0\\\left(1-c\right)\left(1-d\right)\ge0\\\left(1-ab\right)\left(1-cd\right)\ge0\end{cases}}\)

\(\Rightarrow\hept{\begin{cases}a+b\le1+ab\\c+d\le1+cd\\ab+cd\le1+abcd\end{cases}}\)

\(\Rightarrow a+b+c+d\le2+ab+cd\le2+1+abcd=3+abcd\)

Vậy \(VT\le\frac{3+abcd}{1+abcd}\le\frac{3\left(1+abcd\right)}{1+abcd}=3\)

Dấu "=" xảy ra khi a=0,b=c=d=1