FORTY-FOURTH ANNUAL WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION

Saturday, December 3, 1983 Examination A

A-1. How many positive integers n are there such that n is an exact divisor of at least one of the numbers

10^{40}, 20^{30} ?

A-2. The hands of an accurate clock have lengths 3 and 4. Find the distance between the tips of the hands when that distance is increasing most rapidly.

A-3. Let p be in the set {3, 5, 7, 11, …} of odd primes and let

F (n) = 1 + 2n + 3n^{2} + ⋯ + (p − 1)n^{p−2}.

Prove that if a and b are distinct integers in {0, 1, 2, …, p−1} then F (a) and F (b) are not congruent modulo p, that is, F (a) − F (b) is not exactly divisible by p.

A-4. Let k be a positive integer and let m = 6k − 1. Let

S(m) = ∑_{j=1}^{2k−1}(−1)^{j+1}\binom{m}{3j − 1}.

For example with k = 3,

S(17) = \binom{17}{2} − \binom{17}{5} + \binom{17}{8} − \binom{17}{11} + \binom{17}{14}.

Prove that S(m) is never zero. [As usual, \binom{m}{r} = \frac{m!}{r!(m−r)!}.]

A-5. Prove or disprove that there exists a positive real number u such that [u^{n}] − n is an even integer for all positive integers n.

Here [x] denotes the greatest integer less than or equal to x.

A-6. Let exp(t) denote e^{t} and

F (x) = \frac{x^{4}}{exp(x^{3})} ∫_{0}^{x} ∫_{0}^{x−u} exp(u^{3} + v^{3}) dv du.

Find lim_{x→∞} F (x) or prove that it does not exist.

FORTY-FOURTH ANNUAL WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION

Saturday, December 3, 1983 Examination B

B-1. Let v be a vertex (corner) of a cube C with edges of length 4. Let S be the largest sphere that can be inscribed in C. Let R be the region consisting of all points p between S and C such that p is closer to v that to any other vertex of the cube. Find the volume of R.

B-2. For positive integers n, let C(n) be the number of representations of n as a sum of nonincreasing powers of 2, where no power can be used more than three times. For example C(8) = 5 since the representations for 8 are:

8, 4 + 4, 4 + 2 + 2, 4 + 2 + 1 + 1, 2 + 2 + 2 + 1 + 1.

Prove or disprove that there is a polynomial P (x) such that C(n) = [P (n)] for all positive integers n; here [u] denotes the greatest integer less than or equal to u.

B-3. Assume that the differential equation

y′′′ + p(x)y′′ + q(x)y′ + r(x)y = 0

has solutions y_{1}(x), y_{2}(x), and y_{3}(x) on the whole real line such that

y_{1}^{2}(x) + y_{2}^{2}(x) + y_{3}^{2}(x) = 1

for all real x. Let

f(x) = (y′_{1}(x))^{2} + (y′_{2}(x))^{2} + (y′_{3}(x))^{2} .

Find constants A and B such that f(x) is a solution to the differential equation

y′(x) + Ap(x)y = Br(x).

B-4. Let f(n) = n + [\sqrt{n}] where [x] is the largest integer less than or equal to x. Prove that, for every positive integer m, the sequence

m, f(m), f(f(m)), f(f(f(m))), …

contains at least one square of an integer.

B-5. Let ∥u∥ denote the distance from the real number u to the neareast integer. (For example, ∥2.8∥ = .2 = ∥3.2∥.) For positive integers n, let

a_{n} = \frac{1}{n} ∫_{1}^{n} ∥\frac{n}{x}∥ dx.

Determine lim_{n→∞} a_{n}. You may assume the identity

\frac{2}{1} · \frac{2}{3} · \frac{4}{3} · \frac{4}{5} · \frac{6}{5} · \frac{6}{7} · \frac{8}{7} · \frac{8}{9} ⋯ = \frac{π}{2} .

B-6. Let k be a positive integer, let m = 2^{k} + 1, and let r ≠ 1 be a complex root of z^{m} − 1 = 0. Prove that there exists polynomials P (z) and Q(z) with integer coefficients such that

(P (r))^{2} + (Q(r))^{2} = −1.