Inegalitate conditionata

[alex2008]

Daca \( a_i>0\ ,\ i=\overline{1,5} \) si \( a_1+a_2+a_3+a_4+a_5=1 \) , demonstrati ca :

\( (\frac{1}{a_1}-1)(\frac{1}{a_2}-1)...(\frac{1}{a_5}-1)\ge 1024 \) .

[Marius Mainea]

\( LHS=\prod{\frac{1-a_i}{a_i}\ge\prod{\frac{4\sqrt[4]{a_1a_2a_3a_4a_5}}{a_i\sqrt[4]{a_i}}=(2^2)^5=1024 \)