28th Putnam 1967


A1. We are given a positive integer n and real numbers a_i such that \|∑₁ⁿ a_k sin kx\| ≤ \|sin x\| for all real x. Prove \|∑₁ⁿ k a_k\| ≤ 1.
A2. Let u_n be the number of symmetric n x n matrices whose elements are all 0 or 1, with exactly one 1 in each row. Take u₀ = 1. Prove u_n+1 = u_n + n u_n-1 and ∑₀^∞ u_n xⁿ/n! = e^f(x), where f(x) = x + (1/2) x².
A3. Find the smallest positive integer n such that we can find a polynomial nx² + ax + b with integer coefficients and two distinct roots in the interval (0, 1).
A4. Let 1/2 < α ∈ R, the reals. Show that there is no function f : [0, 1] → R such that f(x) = 1 + α ∫_x¹ f(t) f(t - x) dt for all x ∈ [0, 1].
A5. K is a convex, finite or infinite, region of the plane, whose boundary is a union of a finite number of straight line segments. Its area is at least π/4. Show that we can find points P, Q in K such that PQ = 1.
A6. a_i and b_i are reals such that a₁b₂ ≠ a₂b₁. What is the maximum number of possible 4-tuples (sign x₁, sign x₂, sign x₃, sign x₄) for which all x_i are non-zero and x_i is a simultaneous solution of a₁x₁ + a₂x₂ + a₃x₃ + a₄x₄ = 0 and b₁x₁ + b₂x₂ + b₃x₃ + b₄x₄ = 0. Find necessary and sufficient conditions on a_i and b_i for this maximum to be achieved.
B1. A hexagon is inscribed in a circle radius 1. Alternate sides have length 1. Show that the midpoints of the other three sides form an equilateral triangle.
B2. α, β ∈ [0, 1] and we have ax² + bxy + cy² ≡ (αx + (1 - α)y)², (αx + (1 - α)y)(βx + (1 - β)y) ≡ dx² + exy + fy². Show that at least one of a, b, c ≥ 4/9 and at least one of d, e, f ≥ 4/9.
B3. R is the reals. f, g are continuous functions R → R with period 1. Show that lim_n→∞ ∫₀¹ f(x) g(nx) dx = (∫₀¹ f(x) dx) (∫₀¹ g(x) dx).
B4. We are given a sequence a₁, a₂, ... , a_n. Each a_i can take the values 0 or 1. Initially, all a_i = 0. We now successively carry out steps 1, 2, ... , n. At step m we change the value of a_i for those i which are a multiple of m. Show that after step n, a_i = 1 iff i is a square. Devise a similar scheme to give a_i = 1 iff i is twice a square.
B5. The first n terms of the exansion of (2 - 1)^-n are 2^-n ( 1 + n/1! (1/2) + n(n + 1)/2! (1/2)² + ... + n(n + 1) ... (2n -2)/(n - 1)! (1/2)^n-1 ). Show that they sum to 1/2.
B6. R is the reals. D is the closed unit disk x² + y² ≤ 1 in R². The function f : D → [-1, 1] has partial derivatives f₁(x, y) and f₂(x, y). Show that there is a point (a, b) in the interior of D such that f₁(a, b)² + f₂(a, b)² ≤ 16.

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 75 (1968) 734-9. 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 in some university libraries.