OBM 2002

Show that there is a set of $2002$ distinct positive integers such that the sum of one or more elements of the set is never a square, cube, or higher power.

$ABCD$ is a cyclic quadrilateral and $M$ a point on the side $CD$ such that $ADM$ and $ABCM$ have the same area and the same perimeter. Show that two sides of $ABCD$ have the same length.

The squares of an $m\times n$ board are labeled from $1$ to $mn$ so that the squares labeled $i$ and $i+1$ always have a side in common. Show that for some $k$ the squares $k$ and $k+3$ have a side in common.

For any non-empty subset $A$ of $\{1, 2, \ldots , n\}$ define $f(A)$ as the largest element of $A$ minus the smallest element of $A$ . Find $\sum f(A)$ where the sum is taken over all non-empty subsets of $\{1, 2, \ldots , n\}$ .

A finite collection of squares has total area $4$ . Show that they can be arranged to cover a square of side $1$ .

Show that we cannot form more than $4096$ binary sequences of length $24$ so that any two differ in at least $8$ positions.