Solutions should be submitted to

Prof. E.J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3

no later than November 30, 2000.

37.

Let ABC be a triangle with sides a, b, c, inradius r and circumradius R (using the conventional notation). Prove that

38.

Let us say that a set S of nonnegative real numbers if hunky-dory if and only if, for all x and y in S, either x + y or |x - y | is in S. For instance, if r is positive and n is a natural number, then S(n, r) = { 0, r, 2r, ¼, nr } is hunky-dory. Show that every hunky-dory set with finitely many elements is { 0 }, is of the form S(n, r) or has exactly four elements.

39.

(a) ABCDEF is a convex hexagon, each of whose diagonals AD, BE and CF pass through a common point. Must each of these diagonals bisect the area?

40.

Determine all solutions in integer pairs (x, y) to the diophantine equation x² = 1 + 4y³(y + 2).

41.

Determine the least positive number p for which there exists a positive number q such that

42.

G is a connected graph; that is, it consists of a number of vertices, some pairs of which are joined by edges, and, for any two vertices, one can travel from one to another along a chain of edges. We call two vertices adjacent if and only if they are endpoints of the same edge. Suppose there is associated with each vertex v a nonnegative integer f(v) such that all of the following hold:

(2) If f(v) > 0, then v is adjacent to at least one vertex w such that f(w) < f(v).

(3) There is exactly one vertex u such that f(u) = 0.

Prove that f(v) is the number of edges in the chain with the fewest edges connecting u and v.