22nd Putnam 1961


A1. The set of pairs of positive reals (x, y) such that x^y = y^x form the straight line y = x and a curve. Find the point at which the curve cuts the line.
A2. For which real numbers α, β can we find a constant k such that x^αy^β < k(x + y) for all positive x, y?
A3. Find lim_n→∞ ∑₁^N n/(N + i²), where N = n².
A4. If n = ∏p^r be the prime factorization of n, let f(n) = (-1)^{∑ r} and let F(n) = ∑_d\|n f(d). Show that F(n) = 0 or 1. For which n is F(n) = 1?
A5. Let X be a set of n points. Let P be a set of subsets of X, such that if A, B ∈ P, then X - A, A ∪ B, A ∩ B ∈ P. What are the possible values for the number of elements of P?
A6. Consider polynomials in one variable over the finite field F₂ with 2 elements. Show that if n + 1 is not prime, then 1 + x + x² + ... + xⁿ is reducible. Can it be reducible if n + 1 is prime?
A7. S is a non-empty closed subset of the plane. The disk (a circle and its interior) D ⊇ S and if any disk D' ⊇ S, then D' ⊇ D. Show that if P belongs to the interior of D, then we can find two distinct points Q, R ∈ S such that P is the midpoint of QR.
B1. a_n is a sequence of positive reals. h = lim (a₁ + a₂ + ... + a_n)/n and k = lim (1/a₁ + 1/a₂ + ... + 1/a_n)/n exist. Show that h k ≥ 1.
B2. Two points are selected independently and at random from a segment length β. What is the probability that they are at least a distance α (< β) apart?
B3. A, B, C, D lie in a plane. No three are collinear and the four points do not lie on a circle. Show that one point lies inside the circle through the other three.
B4. Given x₁, x₂, ... , x_n ∈ [0, 1], let s = ∑_1≤i<j≤n \|x_i - x_j\|. Find f(n), the maximum value of s over all possible {x_i}.
B5. Let n be an integer greater than 2. Define the sequence a_m by a₁ = n, a_m+1 = n to the power of a_m. Either show that a_m < n!! ... ! (where the factorial is taken m times), or show that a_m > n!! ... ! (where the factorial is taken m-1 times).
B6. Let y be the solution of the differential equation y'' = - (1 + √x) y such that y(0) = 1, y'(0) = 0. Show that y has exactly one zero for x ∈ (0, π/2) and find a positive lower bound for it.
B7. The sequence of non-negative reals satisfies a_n+m ≤ a_na_m for all m, n. Show that lim a_n^1/n exists.

To avoid possible copyright problems, I have changed the wording, but not the substance, of all the problems. The original text and solutions are available in: A M Gleason, R E Greenwood & L M Kelly, The William Lowell Putnam Mathematical Competition, Problems and Solutions, 1938-1964, MAA 1980. Out of print, but available in some university libraries.