Inegalitate simpla

[alex2008]

Fie a, b, c patru numere strict pozitive si distincte . Demonstrati ca : \( \frac{a}{4a+b+c}+\frac{b}{a+4b+c}+\frac{c}{a+b+4c}<\frac{1}{2} \).

[Marius Mainea]

Notam 4a+b+c=x , 4b+c+a=y , 4c+a+b=z de unde rezulta \( a=\frac{5x-y-z}{18} \) si analoagele , si inegalitatea se reduce la

\( \sum{\frac{5x-y-z}{18x}<\frac{1}{2} \)