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\def\pr#1.{\smallskip{\bf Problem #1.}\par}

\pr1.
Let $P$ be a convex polygon with center of gravity $C$.  For each edge
$E$ of $P$, let $L_E$ be the line of which $E$ is a segment.  Prove
that, for at least one edge $E$ of $P$, the foot of the perpendicular
from $C$ to $L_E$ is on $E$.

\pr2.
Suppose that a bag contains balls of $n$ different colors, there being
precisely one ball of the first color, two of the second, and so on,
up to $n$ balls of the $n$th color.  Balls are drawn out of the bag at
random into pockets of sizes 1, 2, \dots, $n$ in order, so that the
first ball drawn is put in the first pocket, the next two in the
second pocket, and so on.  Find the probability $p_n$ that no pocket
contains two balls of the same color. 

\pr3.
Let $a$ be a fixed positive real number.  For what values of $u_0$
does the sequence defined inductively by $u_{n+1}=2u_n-au_n^2$
converge to $1/a$?

\pr4.
A sequence $(a_n)$ is said to {\sl satisfy a linear recurrence of
order\/} $k$ if each term $a_n$ after the first $k$ is expressible as
a linear combination of the $k$ immediately preceding terms, where the
$k$ coefficients of the linear expression do not depend on $n$.  Show
that the sequence 1,~$-1$,~2,~$-2$, 3, $-3$, \dots\ satisfies a linear
recurrence of order 3 but none of lower order.

\pr5.
Find all real numbers $b>0$ such that $x^b\leq b^x$ for all $x>0$.  

\pr6.
Assume that $(a_n)$ is a sequence of real numbers such that, for every
real number $t$, the sequence of fractional parts $\{ta_n\}$ converges
to 0.  (The {\sl fractional part\/} $\{x\}$ of a number $x$ is $x$
minus the greatest integer $\leq x$.)  Prove that $a_n$ converges to
0.  

\pr7.
A {\sl rearrangement\/} of an infinite series $\sum_{n=1}^\infty a_n$
is another series with the same terms in a different order, i.e.,
$\sum_{n=1}^\infty a_{h(n)}$ where $h$ is a permutation of the set of
positive integers.  It is a {\sl mild\/} rearrangement if there is a
constant $M$ such that $|h(n)-n|\leq M$ for all $n$.

(a) [2 points]\quad
Give an example of a convergent series and a divergent rearrangement
of it.

(b) [3 points]\quad
Prove that all mild rearrangements of convergent series converge.

(c) [5 points]\quad
What happens in (b) if the definition of ``mild'' is weakened to
require only $|h(n)-n|\leq\ln n$ for all $n$?  

\bye