FORTY-THIRD ANNUAL WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION

Saturday, December 4, 1982 Examination A

A-1. Let V be the region in the cartesian plane consisting of all points (x, y) satisfying the simultaneous equations

|x| ≤ y ≤ |x| + 3 and y ≤ 4.

Find the centroid (\overline{x}, \overline{y}) of V.

A-2. For positive real x. let

B_{n}(x) = 1^{x} + 2^{x} + 3^{x} + ⋯ + n^{x}.

Prove or disprove the convergence of

∑_{n=2}^{∞} \frac{B_{n}(log_{n} 2)}{(n log_{2} n)^{2}}.

A-3. Evaluate

∫_{0}^{∞} \frac{Arctan(πx) − Arctan(x)}{x} dx.

A-4. Assume that the system of simultaneous differential equations

y′ = −z^{3}, z′ = y^{3}

with the initial conditions y(0) = 1, z(0) = 0 has a unique solution y = f(x), z = g(x) defined for all real x. Prove that there exists a positive constant L such that for all real x,

f(x + L) = f(x), g(x + L) = g(x).

A-5. Let a, b, c, and d be positive integers and

r = 1 − \frac{a}{b} − \frac{c}{d}.

Given that a + c ≤ 1982 and r > 0, prove that

r > \frac{1}{1983^{3}} .

A-6. Let σ be a bijection of the positive integers, that is, a one-to-one function from {1, 2, 3, …} onto itself. Let x_{1}, x_{2}, x_{3}, … be a sequence of real numbers with the following properties:

(i) |x_{n}| is a strictly decreasing function of n; (ii) |σ(n) − n| · |x_{n}| → 0 as n → ∞; (iii) lim_{n→∞} ∑_{k=1}^{n} x_{k} = 1.

Prove or disprove that these conditions imply that

lim_{n→∞} ∑_{k=1}^{n} x_{σ(k)} = 1.

FORTY-THIRD ANNUAL WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION

Saturday, December 4, 1982 Examination B

B-1. Let M be the midpoint of side BC of a general △ABC. Using the smallest possible n, describe a method for cutting △AMB into n triangles which can be reassembled to form a triangle congruent to △AMC.

B-2. Let A(x, y) denote the number of points (m, n) in the plane with integer coordinates m and n satisfying m^{2} + n^{2} ≤ x^{2} + y^{2}. Let g = ∑_{k=0}^{∞} e^{−k^{2}}. Express

∫_{−∞}^{∞} ∫_{−∞}^{∞} A(x, y)e^{−x^{2}−y^{2}} dx dy

as a polynomial in g.

B-3. Let p_{n} be the probability that c + d is a perfect square when the integers c and d are selected independently at random from the set \{1, 2, 3, …, n\}. Show that lim_{n→∞}(p_{n}\sqrt{n}) exists and express this limit in the form r(\sqrt{s}−t), where s and t are integers and r is a rational number.

B-4. Let n_{1}, n_{2}, … n_{s} be distinct integers such that

(n_{1} + k)(n_{2} + k) ⋯ (n_{s} + k)

is an integral multiple of n_{1}n_{2} ⋯ n_{s} for every integer k. For each of the following assertions, give a proof or a counterexample:

(a) |n_{i}| = 1 for some i.

(b) If further all n_{i} are positive, then

(n_{1}, n_{2}, …, n_{s}) = (1, 2, …, s).

B-5. For each x > e^{e} define a sequence S_{x} = u_{0}, u_{1}, u_{2}, … recursively as follows: u_{0} = e, while for n ≥ 0, u_{n+1} is the logarithm of x to the base u_{n}. Prove that S_{x} converges to a number g(x) and that the function g defined in this way is continuous for x > e^{e}.

B-6. Let K(x, y, z) denote the area of a triangle whose sides have lengths x, y, and z. For any two triangles with sides a, b, c and a′, b′, c′, respectively, prove that

\sqrt{K(a, b, c)} + \sqrt{K(a′, b′, c′)}|le\sqrt{K(a + a′, b + b′, c + c′)}

and determine the cases of equality.