30th Putnam 1969


A1. R² represents the usual plane (x, y) with -∞ < x, y < ∞. p: R² → R is a polynomial with real coefficients. What are the possibilities for the image p(R²)?
A2. A is an n x n matrix with elements a_ij = \|i - j\|. Show that the determinant \|A\| = (-1)^n-1 (n - 1) 2^n-2.
A3. An n-gon (which is not self-intersecting) is triangulated using m interior vertices. In other words, there is a set of N triangles such that: (1) their union is the original n-gon; (2) the union of their vertices is the set consisting of the n vertices of the n-gon and the m interior vertices; (3) the intersection of any two distinct triangles in the set is either empty, a vertex of both triangles, or a side of both triangles. What is N?
A4. Prove that ∫₀¹ x^x dx = 1 - 1/2² + 1/3³ - 1/4⁴ + ... .
A5. u(t) is a continuous function. x(t), y(t) is the solution of x' = -2y + u(t), y' = -2x + u(t) satisfying the initial condition x(0) = x₀, y(0) = y₀. Show that if x₀ ≠ y₀, then we do not have x(t) = y(t) = 0 for any t, but that given any x₀ = y₀ and any T > 0, we can always find some u(t) such that x(T) = y(T) = 0.
A6. The sequence a₁ + 2a₂, a₂ + 2a₃, a₃ + 2a₄, ... converges. Prove that the sequence a₁, a₂, a₃, ... also converges.
B1. The positive integer n is divisible by 24. Show that the sum of all the positive divisors of n - 1 (including 1 and n - 1) is also divisible by 24.
B2. G is a finite group with identity 1. Show that we cannot find two proper subgroups A and B (≠ {1} or G) such that A ∪ B = G. Can we find three proper subgroups A, B, C such that A ∪ B ∪ C = G?
B3. The sequence a₁, a₂, a₃, ... satisfies a₁a₂ = 1, a₂a₃ = 2, a₃a₄ = 3, a₄a₅ = 4, ... . Also, lim_n→∞a_n/a_n+1 = 1. Prove that a₁ = √(2/π).
B4. Γ is a plane curve of length 1. Show that we can find a closed rectangle area 1/4 which covers Γ.
B5. The sequence a_i, i = 1, 2, 3, ... is strictly monotonic increasing and the sum of its inverses converges. Let f(x) = the largest i such that a_i < x. Prove that f(x)/x → 0 as x → ∞.
B6. M is a 3 x 2 matrix, N is a 2 x 3 matrix. MN = 8 2 -2 2 5 4 -2 4 5 Show that NM = 9 0 0 9

To avoid possible copyright problems, I have changed the wording, but not the substance, of all the problems. The original text and official solutions were published in American Mathematical Monthly 77 (1970) 723-8. They are also available (with the solutions expanded) in: Gerald L Alexanderson et al, The William Lowell Putnam Mathematical Competition, 1965-1984. Out of print, but available in some university libraries.