FORTY-FOURTH ANNUAL WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION

Saturday, December 5, 1981 Examination A

A-1. Let E(n) denote the largest integer k such that 5^{k} is an integral divisor of the product 1^{1}2^{2}3^{3} ⋯ n^{n}. Calculate

lim_{n→∞} \frac{E(n)}{n}.

A-2. Two distinct squares of the 8 by 8 chessboard C are said to be adjacent if they have a vertex or side in common. Also, g is called a C-gap if for every numbering of C with all the integers 1, 2, …, 64 there exist two adjacent squares whose numbers differ by at least g. Determine the largest C-gap g.

A-3. Find

lim_{t→∞} [e^{−t} ∫_{0}^{t} ∫_{0}^{t} \frac{e^{x} − e^{y}}{x − y} dx dy]

or show that the limit does not exist.

A-4. A point P moves inside a unit square in a straight line at unit speed. When it meets a corner it escapes. When it meets an edge its line of motion is reflected so that the angle of incidence equals the angle of reflection.

Let N(T) be the number of starting directions from a fixed interior point P_{0} for which P escapes within T units of time. Find the least constant a for which constants b and c exist such that

N(T) ≤ aT^{2} + bT + c

for all T > 0 and all initial points P_{0}.

A-5. Let P (x) be a polynomial with real coefficients and form the polynomial

Q(x) = (x^{2} + 1)P (x)P′(x) + x ([P (x)]^{2} + [P′(x)]^{2}) .

Given that the equation P (x) = 0 has n distinct real roots exceeding 1, prove or disprove that the equation Q(x) = 0 has at least 2n − 1 distinct real roots.

A-6. Suppose that each of the vertices of △ABC is a lattice point in the (x, y)-plane and that there is exactly one lattice point P in the interior of the triangle. The line AP is extended to meet BC at E. Determine the largest possible value for the ratio of lengths of segments

\frac{|AP |}{|P E|}.

[A lattice point is a point whose coordinates x and y are integers.]

FORTY-FOURTH ANNUAL WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION

Saturday, December 5, 1981 Examination B

B-1. Find

lim_{n→∞} [\frac{1}{n^{5}} ∑_{h=1}^{n} ∑_{k=1}^{n}(5h^{4} − 18h^{2}k^{2} + 5k^{4})] .

B-2. Determine the minimum value of

(r − 1)^{2} + (\frac{s}{r} − 1)^{2} + (\frac{t}{s} − 1)^{2} (\frac{4}{t} − 1)^{2}

for all real numbers r, s, t with 1 ≤ r ≤ s ≤ t ≤ 4.

B-3. Prove that there are infinitely many positive integers n with the property that if p is a prime divisor of n^{2} + 3, then p is also a divisor of k^{2} + 3 for some integer k with k^{2} < n.

B-4. Let V be a set of 5 by 7 matrices, with real entries and with the property that rA + sB ∈ V whenever A, B ∈ V and r and s are scalars (i.e., real numbers). Prove or disprove the following assertion: If V contains matrices of ranks 0, 1, 2, 4, and 5, then it also contains a matrix of rank 3.

[The rank of a nonzero matrix M is the largest k such that the entries of some k rows and some k columns form a k by k matrix with nonzero determinant.]

B-5. Let B(n) be the numbers of ones in the base two expression for the positive integer n. For example, B(6) = B(110_{2}) = 2 and B(15) = B(1111_{2}) = 4. Determine whether or not

exp (∑_{n=1}^{∞} \frac{B(n)}{n(n + 1)})

is a rational number. Here exp(x) denotes e^{x}.

B-6. Let C be a fixed unit circle in the Cartesian plane. For any convex polygon P each of whose sides is tangent to C, let N(P, h, k) be the number of points common to P and the unit cirle with center (h, k). Let H(P) be the region of all points (x, y) for which N(P, x, y) ≥ 1 and F (P) be the area of H(P). Find the smallest number u with

\frac{1}{F (P)} ∫ ∫ N(P, x, y) dx dy < u

for all polygons P, where the double integral is taken over H(P).