22nd Putnam 1961

A1.  The set of pairs of positive reals (x, y) such that xy = yx 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 xαyβ < k(x + y) for all positive x, y?
A3.  Find limn→∞1N n/(N + i2), where N = n2.
A4.  If n = ∏pr 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.  an is a sequence of positive reals. h = lim (a1 + a2 + ... + an)/n and k = lim (1/a1 + 1/a2 + ... + 1/an)/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, ... , xn ∈ [0, 1], let s = ∑1≤i<j≤n |xi - xj|. Find f(n), the maximum value of s over all possible {xi}.
B5.  Let n be an integer greater than 2. Define the sequence am by a1 = n, am+1 = n to the power of am. Either show that am < n!! ... ! (where the factorial is taken m times), or show that am > 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 an+m ≤ anam for all m, n. Show that lim an1/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.

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© John Scholes
20 Oct 1999