PROBLEMS FOR DECEMBER

Please send your solution to

Ed Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3

no later than January 15, 2004. It is important that your complete mailing address and your email address appear on the front page.

276.

Let a, b, c be the lengths of the sides of a triangle and let s = ¹/₂(a + b + c) be its semi-perimeter and r be the radius of the inscribed circle. Prove that

(s - a)^-2 + (s - b)^-2 + (s - c)^-2 ³ r^-2

and indicate when equality holds.

277.: Let m and n be positive integers for which m < n. Suppose that an arbitrary set of n integers is given and the following operation is performed: select any m of them and add 1 to each. For which pairs (m, n) is it always possible to modify the given set by performing the operation finitely often to obtain a set for which all the integers are equal?

278.: (a) Show that 4mn - m - n can be an integer square for infinitely many pairs (m, n) of integers. Is it possible for either m or n to be positive?

: (b) Show that there are infinitely many pairs (m, n) of positive integers for which 4mn - m - n is one less than a perfect square.

279.: (a) For which values of n is it possible to construct a sequence of abutting segments in the plane to form a polygon whose side lengths are 1, 2, ¼, n exactly in this order, where two neighbouring segments are perpendicular?

: (b) For which values of n is it possible to construct a sequence of abutting segments in space to form a polygon whose side lengths are 1, 2, ¼, n exactly in this order, where any two of three sucessive segments are perpendicular?

280.: Consider all finite sequences of positive integers whose sum is n. Determine T(n, k), the number of times that the positive integer k occurs in all of these sequences taken together.

281.: Let a be the result of tossing a black die (a number cube whose sides are numbers from 1 to 6 inclusive), and b the result of tossing a white die. What is the probability that there exist real numbers x, y, z for which x + y + z = a and xy + yz + zx = b?

282.: Suppose that at the vertices of a pentagon five integers are specified in such a way that the sum of the integers is positive. If not all the integers are non-negative, we can perform the following operation: suppose that x, y, z are three consecutive integers for which y < 0; we replace them respectively by the integers x + y, -y, z + y. In the event that there is more than one negative integer, there is a choice of how this operation may be performed. Given any choice of integers, and any sequence of operations, must we arrive at a set of nonnegative integers after a finite number of steps?

: For example, if we start with the numbers (2, -3, 3, -6, 7) around the pentagon, we can produce (1, 3, 0, -6, 7) or (2, -3, -3, 6, 1).