JMO 1998

Let $p \geq 3$ be a prime, and let $p$ points $A_{0}, \ldots, A_{p-1}$ lie on a circle in that order. Above the point $A_{1+\cdots+k-1}$ we write the number $k$ for $k=1, \ldots, p$ (so $1$ is written above $A_{0}$). How many points have at least one number written above them?

A country has $1998$ airports connected by some direct flights. For any three airports, some two are not connected by a direct flight. What is the maximum number of direct flights that can be offered?

Let $P_{1}, \ldots P_{n}$ be the sequence of vertices of a closed polygons whose sides may properly intersect each other at points other than the vertices. The external angle at $P_{i}$ is defined as $180^\circ$ minus the angle of rotation about $P_{i}$ required to bring the ray $P_{i}P_{i-1}$ onto the ray $P_{i}P_{i+1}$, taken in the range ($0^\circ, 360^\circ$). (Here $P_{0}=P_{n}$ and $P_{1}=P_{n+1}$). Prove that if the sum of the external angles is a multiple of $720^\circ$, then the number of self-intersections is odd.

Let $c_{n, m}$ be the number of permutations of $\{1, \ldots, n\}$ which can be written as the product of $m$ transpositions of the form $(i, i+1)$ for some $i=1, \ldots, n-1$ but not of $m-1$ suct transpositions. Prove that for all $n \in \mathbb{N}$, $$\sum_{m=0}^\infty c_{n, m}t^{m}= \prod_{i=1}^{n}(1+t+\cdots+t^{i-1}).$$

On each of $12$ points around a circle we place a disk with one white side and one black side. We may perform the following move: select a black disk, and reverse its two neighbors. Find all initial configurations from which some sequence of such moves leads to the position where all disks but one are white.