FORTY-EIGHTH ANNUAL WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION

Saturday, December 5, 1987 Examination A

A-1. Curves A, B, C, and D, are defined in the plane as follows:

\begin{aligned}A & = \{(x, y) : x^{2} − y^{2} = \frac{x}{x^{2} + y^{2}} \}, \\ B & = \{(x, y) : 2xy + \frac{y}{x^{2} + y^{2}} = 3\}, \\ C & = \{(x, y) : x^{3} − 3xy^{2} + 3y = 1\}, \\ D & = \{(x, y) : 3x^{2}y − 3x − y^{3} = 0 \}. \end{aligned}

Prove that A ∩ B = C ∩ D.

A-2. The sequence of digits

1 2 3 4 5 6 7 8 9 1 0 1 1 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 …

is obtained by writing the positive integers in order. If the 10^{n}th digit in this sequence occurs in the part of the sequence in which the m-digit numbers are placed, define f(n) to be m. For example f(2) = 2 because the 100th digit enters the sequence in the placement of the two-digit integer 55. Find, with proof, f(1987).

A-3. For all real x, the real-valued function y = f(x) satisfies

y′′ − 2y′ + y = 2e^{x}.

(a) If f(x) > 0 for all real x, must f′(x) > 0 for all real x? Explain.

(b) If f′(x) > 0 for all real x, must f(x) > 0 for all real x? Explain.

A-4. Let P be a polynomial, with real coefficients, in three variables and F be a function of two variables such that

P (ux, uy, uz) = u^{2}F (y − x, z − x) for all real x, y, z, u,

and such that P (1, 0, 0) = 4, P (0, 1, 0) = 5, and P (0, 0, 1) = 6. Also let A, B, C be complex numbers with P (A, B, C) = 0 and |B − A| = 10. Find |C − A|.

A-5. Let

G(x, y) = (\frac{−y}{x^{2} + 4y^{2}} , \frac{x}{x^{2} + 4y^{2}} , 0).

Prove or disprove that there is a vector-valued function

F(x, y, z) = (M(x, y, z), N(x, y, z), P (x, y, z))

with the following properties:

(i) M, N, P have continuous partial derivatives for all (x, y, z) ≠ (0, 0, 0); (ii) curl F = 0 for all (x, y, z) ≠ (0, 0, 0); (iii) F(x, y, 0) = G(x, y).

A-6. For each positive integer n, let a(n) be the number of zeros in the base 3 representation of n. For which positive real numbers x does the series

∑_{n=1}^{∞} \frac{x^{a(n)}}{n^{3}}

converge?

FORTY-EIGHTH ANNUAL WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION

Saturday, December 5, 1987 Examination B

B-1. Evaluate:

∫_{2}^{4} \frac{\sqrt{ln(9 − x)} dx}{\sqrt{ln(9 − x)} + \sqrt{ln(x + 3)}}.

B-2. Let r, s, and t be integers with 0 ≤ r, 0 ≤ s, and r + s ≤ t. Prove that

\begin{matrix}( & {s \atop 0} & ) \\ ( & {t \atop r} & ) \end{matrix} + \begin{matrix}( & {s \atop 1} & ) \\ ( & {t \atop r+1} & ) \end{matrix} + \begin{matrix}( & {s \atop 2} & ) \\ ( & {t \atop r+2} & ) \end{matrix} + ⋯ + \begin{matrix}( & {s \atop s} & ) \\ ( & {t \atop r+s} & ) \end{matrix} = \frac{t + 1}{(t + 1 − s)\binom{t−s}{r}}.

B-3. Let F be a field in which 1+1 ≠ 0. Show that the set of solutions to the equation x^{2} +y^{2} = 1 with x and y in F is given by (x, y) = (1, 0) and

(x, y) = (\frac{r^{2} − 1}{r^{2} + 1}, \frac{2r}{r^{2} + 1}),

where r runs through the elements of F such that r^{2} ≠ −1.

B-4. Let (x_{1}, y_{1}) = (0.8, 0.6) and let x_{n+1} = x_{n} cos y_{n} − y_{n} sin y_{n} and y_{n+1} = x_{n} sin y_{n} + y_{n} cos y_{n} for n = 1, 2, 3, . . .. For each of lim_{n→∞} x_{n} and lim_{n→∞} y_{n}, prove that the limit exists and find it or prove that the limit does not exist.

B-5. Let O_{n} be the n-dimensional zero vector (0,0,...,0). Let M be a 2n × n matrix of complex numbers such that whenever (z_{1}, z_{2}, …, z_{n})M = O_{n}, with complex z_{i}, not all zero, then at least one of the z_{i} is not real. Prove that for arbitrary real numbers r_{1}, r_{2}, … r_{2n}, there are complex numbers w_{1}, w_{2}, …, w_{n} such that

Re [M(w_{1}, …, w_{n})^{t}] = (r_{1}, …, r_{2n})^{t}.

(Note: If C is a matrix of complex numbers, Re(C) is the matrix whose entries are the real parts of the entries of C.)

B-6. Let F be the field of p^{2} elements where p is an odd prime. Suppose S is a set of (p^{2} − 1)/2 distinct nonzero elements of F with the property that for each a ≠ 0 in F, exactly one of a and −a is in S. Let N be the number of elements in the intersection S ∩ {2a : a ∈ S}. Prove that N is even.