18th Putnam 1958

A1.  Show that the real polynomial ∑0n aixi has at least one real root if ∑ ai/(i + 1) = 0.
A2.  A rough sphere radius R rests on top of a fixed rough sphere radius R. It is displaced slightly and starts to roll off. At what point does it lose contact?
A3.  A sequence of numbers αi ∈ [0, 1] is chosen at random. Show that the expected value of n, where ∑1n αi > 1, ∑1n-1 αi ≤ 1 is e.
A4.  z1, z2, ... , zn are complex numbers with modulus a > 0. Let f(n, m) denote the sum of all products of m of the numbers. For example, f(3, 2) = z1z2 + z2z3 + z3z1. Show that |f(n, m)| /am= |f(n, n-m)| /an-m.
A5.  Let R be the reals. Show that there is at most one continuous function f : [0, 1]2 → R satisfying f(x, y) = 1 + ∫0x0y f(s, t) dt ds.
A6.  Assume that the interest rate is r, so that capital of k becomes k(1 + r)n after n years. How much do we need to invest to be able to withdraw 1 at the end of year 1, 4 at the end of year 2, 9 at the end of year 3, 16 at the end of year 4 and so on (in perpetuity)?
A7.  Show that we cannot place 10 unit squares in the plane so that no two have an interior point in common and one has a point in common with each of the others.
B1.  Do both (1) and (2):

(1)   Given real numbers a, b, c, d with a > b, c, d, show how to construct a quadrilateral with sides a, b, c, d and the side length a parallel to that length b. What conditions must a, b, c, d satisfy?

(2)   H is the foot of the altitude from A in the acute-angled triangle ABC. D is any point on the segment AH. BD meets AC at E, and CD meets AB at F. Show that ∠AHE = ∠AHF.

B2.  Let n be a positive integer. Prove that n(n + 1)(n + 2)(n + 3) cannot be a square or a cube.
B3.  In a tournament of n players, every pair of players plays once. There are no draws. Player i wins wi games. Prove that we can find three players i, j, k such that i beats j, j beats k and k beats i iff ∑ wi2 < (n - 1)n(2n - 1)/6.
B4.  Let S be a spherical shell radius 1. Find the average straight line distance between two points of S. [In other words S is the set of points (x, y, z) with x2 + y2 + z2 = 1).
B5.  S is an infinite set of points in the plane. The distance between any two points of S is integral. Prove that S is a subset of a straight line.
B6.  A particle of unit mass moves in a vertical plane under the influence of constant gravitational force g and a resistive force which is in the opposite direction to its velocity and with magnitude a function of its speed. The particle starts at time t = 0 and has coordinates (x, y) at time t. Given that x = x(t) and is not constant, show that y(t) = - g x(t) ∫0t ds/x'(s) + g ∫0t x(s) / x'(s) ds + a x(t) + b, where a and b are constants.
B7.  R is the reals. f : [a, b] → R is continuous and ∫ab xnf(x) dx = 0 for all non-negative integers n. Show that f(x) = 0 for all x.

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