and 
be positive real numbers such that 
.![\[
\frac{x^{3}}{(1 + y)(1 + z)}+\frac{y^{3}}{(1 + z)(1 + x)}+\frac{z^{3}}{(1 + x)(1 + y)}
\geq \frac{3}{4}.
\]](http://latex.artofproblemsolving.com/5/6/5/56568d27a296f1c900fabbb1042549cd2e552e29.png)
,![\[ \frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1. \]](http://latex.artofproblemsolving.com/b/2/f/b2ff456632d4dd3b9ed93e4cba437bdd34d54cd8.png)
Y by
Several jugs (not necessarily of the same size) with juices are placed along a circle. It is allowed to transfuse any part of juice (maybe nothing or the total content) from any jug to the neighboring one on the right, so that the latter one is not overflowed and the sugariness of its content becomes equal to 
. It is known that at the initial moment such transfusion is possible from each jug. Prove that it is possible to perform several transfusions in some order, at most one transfusion from each jug, such that the sugariness of the content of each non-empty jug will become equal to . (Sugariness is the percent of sugar in a jug, by weight. Sugar is always uniformly distributed in a jug.)
. It is known that at the initial moment such transfusion is possible from each jug. Prove that it is possible to perform several transfusions in some order, at most one transfusion from each jug, such that the sugariness of the content of each non-empty jug will become equal to . (Sugariness is the percent of sugar in a jug, by weight. Sugar is always uniformly distributed in a jug.)
