FORTY-SEVENTH ANNUAL WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION

Saturday, December 6, 1986 Examination A

A-1. Find, with explanation, the maximum value of f(x) = x^{3} − 3x on the set of all real numbers x satisfying x^{4} + 36 ≤ 13x^{2}.

A-2. What is the units (i.e., rightmost) digit of [\frac{10^{20000}}{10^{100} + 3}]? Here [x] is the greatest

integer ≤ x.

A-3. Evaluate ∑_{n=0}^{∞} Arccot(n^{2} + n + 1), where Arccot t for t ≥ 0 denotes the number θ in the interval 0 < θ ≤ π/2 with cot θ = t.

A-4. A transversal of an n × n matrix A consists of n entries of A, no two in the same row or column. Let f(n) be the number of n × n matrices A satisfying the following two conditions:

(a) Each entry α_{i,j} of A is in the set {−1, 0, 1}.

(b) The sum of the n entries of a transversal is the same for all transversals of A. An example of such a matrix A is

A = ( \begin{matrix}−1 & 0 & −1 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \end{matrix} ) .

Determine with proof a formula for f(n) of the form

f(n) = a_{1}b_{1}^{n} + a_{2}b_{2}^{n} + a_{3}b_{3}^{n} + a_{4},

where the a_{i}’s and b_{i}’s are rational numbers.

A-5. Suppose f_{1}(x), f_{2}(x), …, f_{n}(x) are functions of n real variables x = (x_{1}, …, x_{n}) with continuous second-order partial derivatives everywhere on R^{n}. Suppose further that there are constants c_{ij} such that

\frac{∂f_{i}}{∂x_{j}} = \frac{∂f_{j}}{∂x_{i}} = c_{ij}

for all i and j, 1 ≤ i ≤ n, 1 ≤ j ≤ n. Prove that there is a function g(x) on R^{n} such that f_{i} + ∂g/∂x_{i} is linear for all i, 1 ≤ i ≤ n. (A linear function is one of the form

a_{0} + a_{1}x_{1} + a_{2}x_{2} + ⋯ + a_{n}x_{n}.)

A-6. Let a_{1}, a_{2}, …, a_{n} be real numbers, and let b_{1}, b_{2}, …, b_{n} be distinct positive integers. Suppose there is a polynomial f(x) satisfying the identity

(1 − x)^{n}f(x) = 1 + ∑_{i=1}^{n} a_{i}x^{b_{i}}.

Find a simple expression (not involving any sums) for f(1) in terms of b_{1}, b_{2}, …, b_{n} and n (but independent of a_{1}, a_{2}, …, a_{n}).

FORTY-SEVENTH ANNUAL WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION

Saturday, December 6, 1986 Examination B

B-1. Inscribe a rectangle of base b and height h and an isosceles triangle of base b in a circle of radius one as shown. For what value of h do the rectangle and triangle have the same area?

h  b

B-2. Prove that there are only a finite number of possibilities for the ordered triple T = (x−y, y −z, z −x) where x, y, and z are complex numbers satisfying the simultaneous equations

x(x − 1) + 2yz = y(y − 1) + 2zx = z(z − 1) + 2xy,

and list all such triples T.

B-3. Let Γ consist of all polynomials in x with integer coefficients. For f and g in Γ and m a positive integer, let f ≡ g (mod m) mean that every coefficient of f −g is an integral multiple of m. Let n and p be positive integers with p prime. Given that f, g, h, r, and s are in Γ with rf + sg ≡ 1 (mod p) and fg ≡ h (mod p), prove that there exist F and G in Γ with F ≡ f (mod p), G ≡ g (mod p), and F G ≡ h (mod p^{n}).

B-4. For a positive real number r, let G(r) be the minimum value of |r − \sqrt{m^{2} + 2n^{2}}| for all integers m and n. Prove or disprove the assertion that lim_{r→∞} G(r) exists and equals 0.

B-5. Let f(x, y, z) = x^{2} + y^{2} + z^{2} + xyz. Let p(x, y, z), q(x, y, z), r(x, y, z) be polynomials with real coefficients satisfying

f(p(x, y, z), q(x, y, z), r(x, y, z)) = f(x, y, z).

Prove or disprove the assertion that the sequence p, q, r consists of some permutation of ±x, ±y, ±z, where the number of minus signs is 0 or 2.

B-6. Suppose A, B, C, D are n × n matrices with entries in a field F, satisfying the conditions that AB^{t} and CD^{t} are symmetric and AD^{t} − BC^{t} = I. Here I is the n × n identity matrix, and if M is an n × n matrix, M^{t} is the transpose of M. Prove that A^{t}D − C^{t}B = I.