SEVENTY-FIRST ANNUAL WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION

Saturday, December 4, 2010 Examination A

A1. Given a positive integer n, what is the largest k such that the numbers 1, 2, …, n can be put into k boxes so that the sum of the numbers in each box is the same? [When n = 8, the example {1, 2, 3, 6}, {4, 8}, {5, 7} shows that the largest k is at least 3.]

A2. Find all differentiable functions f : R → R such that

f′(x) = \frac{f(x + n) − f(x)}{n}

for all real numbers x and all positive integers n.

A3. Suppose that the function h : R^{2} → R has continuous partial derivatives and satisfies the equation

h(x, y) = a\frac{∂h}{∂x}(x, y) + b\frac{∂h}{∂y}(x, y)

for some constants a, b. Prove that if there is a constant M such that |h(x, y)| ≤ M for all (x, y) ∈ R^{2}, then h is identically zero.

A4. Prove that for each positive integer n, the number 10^{10^{10^{n}}} + 10^{10^{n}} + 10^{n} − 1 is not prime.

A5. Let G be a group, with operation ∗. Suppose that

(i) G is a subset of R^{3} (but ∗ need not be related to addition of vectors); (ii) For each a, b ∈ G, either a × b = a ∗ b or a × b = 0 (or both), where × is the usual

cross product in R^{3}. Prove that a × b = 0 for all a, b ∈ G.

A6. Let f : [0, ∞) → R be a strictly decreasing continuous function such that lim_{x→∞} f(x) = 0. Prove that ∫_{0}^{∞} \frac{f(x)−f(x+1)}{f(x)} dx diverges.

SEVENTY-FIRST ANNUAL WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION

Saturday, December 4, 2010 Examination B

B1. Is there an infinite sequence of real numbers a_{1}, a_{2}, a_{3}, … such that

a_{1}^{m} + a_{2}^{m} + a_{3}^{m} + ⋯ = m

for every positive integer m?

B2. Given that A, B, and C are noncollinear points in the plane with integer coordinates such that the distances AB, AC, and BC are integers, what is the smallest possible value of AB?

B3. There are 2010 boxes labeled B_{1}, B_{2}, …, B_{2010}, and 2010n balls have been distributed among them, for some positive integer n. You may redistribute the balls by a sequence of moves, each of which consists of choosing an i and moving exactly i balls from box B_{i} into any one other box. For which values of n is it possible to reach the distribution with exactly n balls in each box, regardless of the initial distribution of balls?

B4. Find all pairs of polynomials p(x) and q(x) with real coefficients for which

p(x)q(x + 1) − p(x + 1)q(x) = 1.

B5. Is there a strictly increasing function f : R → R such that f′(x) = f(f(x)) for all x?

B6. Let A be an n × n matrix of real numbers for some n ≥ 1. For each positive integer k, let A^{[k]} be the matrix obtained by raising each entry to the k^{th} power. Show that if A^{k} = A^{[k]} for k = 1, 2, …, n + 1, then A^{k} = A^{[k]} for all k ≥ 1.