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\centerline{\bf AMS Problem E3377}
\bigskip
In a certain country, the ministry of agriculture has divided the land into as
many farms of equal size as there are families.  Working independently, the
ministry of mining has also divided the land into the same number of parcels
bearing subsurface mineral rights, also of equal size.  The ministry
of assignments discovers that it can allot the farms and minerals to 
families in such a way that each family's two parcels overlap.  In fact,
the overlaps are large enough to allow room for a mine shaft.  Is this a
remarkable coincidence?

Let $n$ be the number of families, and let the unit of area be the size
of one parcel of land.
\medskip
\item{(1.)} Show that there is a positive lower bound $\delta↓n$ depending on
$n$, for which the ministry of assignments can achieve overlap by an area
of at least $\delta↓n$, no matter how the ministries of agriculture and mining
have divided the land.

\item{(2.)} Determine the best (largest) possible value of $\delta↓n$.
\bye