15th Putnam 1955

A1.  Prove that if a, b, c are integers and a√2 + b√3 + c = 0, then a = b = c = 0.
A2.  O is the center of a regular n-gon P1P2 ... Pn and X is a point outside the n-gon on the line OP1. Show that XP1 XP2 ... XPn + OP1n = OXn.
A3.  an is a sequence of monotonically decreasing positive terms such that ∑ an converges. S is the set of all ∑ bn, where bn is a subsequence of an. Show that S is an interval iff an-1 ≤ ∑n ai for all n.
A4.  n vertices are taken on a circle and all possible chords are drawn. No three chords are concurrent (except at a vertex). How many points of intersection are there (excluding vertices)?
A5.  Given a parabola, construct the focus (with ruler and compass).
A6.  For what positive integers n does the polynomial p(x) = xn + (2 + x)n + (2 - x)n have a rational root.
A7.  k is a real constant. y satisfies y'' = (x3 + kx) y with initial conditions y = 1, y' = 0 at x = 0. Show that the solutions of y = 0 are bounded above but not below.
B1.  The lines L and M are horizontal and intersect at O. A sphere rolls along supported by L and M. What is the locus of its center?
B2.  Let R be the reals. f : R → R is twice differentiable, f '' is continuous and f(0) = 0. Define g : R → R by g(x) = f(x)/x for x ≠ 0, g(0) = f '(0). Show that g is differentiable and that g' is continuous.
B3.  Let S be a spherical cap with distance taken along great circles. Show that we cannot find a distance preserving map from S to the plane.
B4.  Can we find n such that μ(n) = μ(n + 1) = ... = μ(n + 1000000) = 0? [The Möbius function μ(r) = 0 iff r has a square factor > 1.]
B5.  n is a positive integer. An infinite sequence of 0s and 1s is such that it only contains n different blocks of n consecutive terms. Show that it is eventually periodic.
B6.  Let N be the set of positive integers and R+ the positive reals. f : N → R+ satisfies f(n) → 0 as n → ∞. Show that there are only finitely many solutions to f(a) + f(b) + f(c) = 1.
B7.  A three-dimensional solid acted on by four constant forces is in equilibrium. No two lines of force are in the same plane. Show that the four lines of force are rulings on a hyperboloid.

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