45th Putnam 1984


A1. S is an a x b x c brick. T is the set of points a distance 1 or less from S. Find the volume of T.
A2. Evaluate 6/( (9 - 4)(3 - 2) ) + 36/( (27 - 8)(9 - 4) ) + ... + 6ⁿ/( (3ⁿ⁺¹ - 2ⁿ⁺¹)(3ⁿ - 2ⁿ) ) + ... .
A3. Let A be the 2n x 2n matrix whose diagonal elements are all x and whose off-diagonal elements a_ij = a for i + j even, and b for i + j odd. Find lim_x→adet A/(x - a)^2n-2.
A4. A convex pentagon inscribed in a circle radius 1 has two perpendicular diagonals which intersect inside the pentagon. What is the maximum area the pentagon can have?
A5. V is the pyramidal region x, y, z ≥ 0, x + y + z ≤ 1. Evaluate ∫_V x y⁹ z⁸ (1 - x - y - z)⁴ dx dy dz.
A6. Let f(n) be the last non-zero digit in the decimal representation of n! . Show that for distinct integers a_i ≥ 0, f(5^a₁ + 5^a₂ + ... + 5^a_r) depends only on a₁ + ... + a_r = a. Write the value as g(a). Find the smallest period for g, or show that it is not periodic.
B1. Define f(n) = 1! + 2! + ... + n! . Find a recurrence relation f(n + 2) = a(n) f(n + 1) + b(n) f(n), where a(x) and b(x) are polynomials.
B2. Find the minimum of f(x, y) = (x - y)² + ( √(2 - x²) - 9/y )² in the half-infinite strip 0 < x < √2, y > 0.
B3. Let S_n be a set with n elements. Can we find a binary operation * on S which satisfies (1) right cancellation: ac = bc implies a = b (for all a, b, c), and (2) total non-associativity: a * (b * c) ≠ (a * b) * c for all a, b, c? Note that we are not just requiring that * is not associative, but that it is never associative.
B4. Find all real valued functions f(x) defined on [0, ∞), such that (1) f is continuous on [0, ∞), (2) f(x) > 0 for x > 0, (3) for all x₀ > 0, the centroid of the region under the curve y = f(x) between 0 and x₀ has y-coordinate equal to the average value of f(x) on [0, x₀].
B5. Let f(n) be the number of 1s in the binary expression for n. Let g(m) = ±0^m ±1^m ±2^m ... ±(2^m - 1)^m, where we take the + sign for k^m iff f(k) is even. Show that g(m) can be written in the form (-1)^m a^p(m) (q(m))! where a is an integer and p(x) and q(x) are polynomials.
B6. Define a sequence of convex polygons P_n as follows. P₀ is an equilateral triangle side 1. P_n+1 is obtained from P_n by cutting off the corners one-third of the way along each side (for example P₁ is a regular hexagon side 1/3). Find lim_n→∞area(P_n).

To avoid possible copyright problems, I have changed the wording, but not the substance, of all the problems. The original text and official solutions were published in American Mathematical Monthly 92 (1985) 561. They are also available (with the solutions expanded) in:

Gerald L Alexanderson et al, The William Lowell Putnam Mathematical Competition, 1965-1984.

Out of print, but available in some university libraries.