17th Putnam 1957

A1.  A surface S in 3-space is such that every normal intersects a fixed line L. Show that we can find a surface of revolution containing S.
A2.  k is a real number greater than 1. A uniform wire consists of the curve y = ex between x = 0 and x = k, and the horizontal line y = ek between x = k - 1 and x = k. The wire is suspended from (k - 1, ek) and a horizontal force applied at the other end, (0, 1) to keep it in equilibrium. Show that the force is directed towards increasing x.
A3.  A and B are real numbers such that cos A ≠ cos B. Show that for any integer n > 1, |cos nA cos B - cos A cos nB| < (n2 - 1) |cos A - cos B|.
A4.  p(z) is a polynomial of degree n with complex coefficients. Its roots (in the complex plane) can be covered by a disk radius r. Show that for any complex k, the roots of n p(z) - k p'(z) can be covered by a disk radius r + |k|.
A5.  Let S be a set of n points in the plane such that the greatest distance between two points of S is 1. Show that at most n pairs of points of S are a distance 1 apart.
A6.  Define an by a1 = ln α, a2 = ln(α - a1), an+1 = an + ln(α - an). Show that limn→∞ an = α - 1.
A7.  Show that we can find a set of disjoint circles such that given any rational point on the x-axis, there is a circle touching the x-axis at that point. Show that we cannot find such a set for the irrational points.
B1.  Let A be the 100 x 100 matrix with amn = mn. Show that the absolute value of each of the 100! products in the expansion of det A is congruent to 1 mod 101.
B2.  The sequence an is defined by its initial value a1, and an+1 = an(2 - k an). For what real a1 does the sequence converge to 1/k?
B3.  R+ is the positive reals, f : [0, 1] → R+ is monotonic decreasing. Show that ∫01 f(x) dx ∫01 x f(x)2 dx ≤ ∫01 x f(x) dx ∫01 f(x)2 dx.
B4.  Show that the number of ways of representing n as an ordered sum of 1s and 2s equals the number of ways of representing n + 2 as an ordered sum of integers > 1. For example: 4 = 1 + 1 + 1 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 2 + 1 = 1 + 1 + 2 (5 ways) and 6 = 4 + 2 = 2 + 4 = 3 + 3 = 2 + 2 + 2 (5 ways).
B5.  Let S be a set and P the set of all subsets of S. f : P → P is such that if X ⊆ Y, then f(X) ⊆ f(Y). Show that for some K, f(K) = K.
B6.  y is the solution of the differential equation (x2 + 9) y'' + (x2 + 4)y = 0, y(0) = 0, y'(0) = 1. Show that y(x) = 0 for some x ∈ [√(63/53)π, 3π/2].
B7.  Let P be a regular polygon and its interior. Show that for any n > 1, we can find a subset Sn of the plane such that we cannot translate and rotate P to cover Sn but we can translate and rotate P to cover any n points of Sn.

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.

Putnam home
© John Scholes
20 Oct 1999