,
,
cut three common chords concurrent at 
.
,
which are internally tangent to all 
.
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
In Chicago, there are 36 criminal gangs, some of which are at war with
each other. Each gangster belongs to several gangs and every pair of gangsters
belongs to a different set of gangs. It is known that no gangster is a member of
two gangs which are at war with each other. Furthermore, each gang that some
gangster does not belong to is at war with some gang he does belong to. What is
the largest possible number of gangsters in Chicago?
each other. Each gangster belongs to several gangs and every pair of gangsters
belongs to a different set of gangs. It is known that no gangster is a member of
two gangs which are at war with each other. Furthermore, each gang that some
gangster does not belong to is at war with some gang he does belong to. What is
the largest possible number of gangsters in Chicago?
.
is the answer, and the construction is in the above reply. It is a result of Moon and Moser (1965). I didn't read their paper myself because it's behind a paywall, but if anyone is interested, google 'Moon, J. W.; Moser, L. (1965), "On cliques in graphs"'.
