PL-OM 1987

There are $n \ge 2$ points in a square side $1$ . Show that one can label the points $P_1, P_2, ... , P_n$ such that $\sum_{i=1}^n |P_{i-1} - P_i|^2 \le 4$ , where we use cyclic subscripts, so that $P_0$ means $P_n$ .

A regular $n$ -gon is inscribed in a circle radius $1$ . Let $X$ be the set of all arcs $PQ$ , where $P, Q$ are distinct vertices of the $n$ -gon. $5$ elements $L_1, L_2, ... , L_5$ of $X$ are chosen at random (so two or more of the $L_i$ can be the same). Show that the expected length of $L_1 \cap L_2 \cap L_3 \cap L_4 \cap L_5$ is independent of $n$ .

$w(x)$ is a polynomial with integer coefficients. Let $p_n$ be the sum of the digits of the number $w(n)$ . Show that some value must occur infinitely often in the sequence $p_1, p_2, p_3, ...$ .

Let $S$ be the set of all tetrahedra which satisfy: (1) the base has area $1$ , (2) the total face area is $4$ , and (3) the angles between the base and the other three faces are all equal. Find the element of $S$ which has the largest volume.

Find the smallest $n$ such that $n^2 -n+11$ is the product of four primes (not necessarily distinct).

A plane is tiled with regular hexagons of side $1$ . $A$ is a fixed hexagon vertex. Find the number of paths $P$ such that: (1) one endpoint of $P$ is $A$ , (2) the other endpoint of $P$ is a hexagon vertex, (3) $P$ lies along hexagon edges, (4) $P$ has length $60$ , and (5) there is no shorter path along hexagon edges from $A$ to the other endpoint of $P$ .