OMM 1995

$N$ students are seated at desks in an $m \times n$ array, where $m, n \ge 3$ . Each student shakes hands with the students who are adjacent horizontally, vertically or diagonally. If there are $1020$ handshakes, what is $N$ ?

Consider 6 points on a plane such that 8 of the distances between them are equal to 1. Prove that there are at least 3 points that form an equilateral triangle.

$A, B, C, D$ are consecutive vertices of a regular $7$ -gon. $AL$ and $AM$ are tangents to the circle center $C$ radius $CB$ . $N$ is the intersection point of $AC$ and $BD$ . Show that $L, M, N$ are collinear.

Find $26$ elements of $\{1, 2, 3, ... , 40\}$ such that the product of two of them is never a square. Show that one cannot find $27$ such elements.

$ABCDE$ is a convex pentagon such that the triangles $ABC, BCD, CDE, DEA$ and $EAB$ have equal areas. Show that $(1/4)$ area $(ABCDE) <$ area $(ABC) < (1/3)$ area $(ABCDE)$ .

A $1$ or $0$ is placed on each square of a $4 \times 4$ board. One is allowed to change each symbol in a row, or change each symbol in a column, or change each symbol in a diagonal (there are $14$ diagonals of lengths $1$ to $4$ ). For which arrangements can one make changes which end up with all $0$ s?