9th Mexican 1995


A1. N students are seated at desks in an m x n array, where m, n ≥ 3. Each student shakes hands with the students who are adjacent horizontally, vertically or diagonally. If there are 1020 handshakes, what is N?
A2. 6 points in the plane have the property that 8 of the distances between them are 1. Show that three of the points form an equilateral triangle with side 1.
A3. 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 point of intersection of AC and BD. Show that L, M, N are collinear.
B1. 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.
B2. ABCDE is a convex pentagon such that the triangles ABC, BCD, CDE, DEA and EAB have equal area. Show that (1/4) area ABCDE < area ABC < (1/3) area ABCDE.
B3. A 1 or 0 is placed on each square of a 4 x 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 0s?

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.