6th Putnam 1946

A1.  p(x) is a real polynomial of degree less than 3 and satisfies |p(x)| ≤ 1 for x ∈ [-1, 1]. Show that |p'(x)| ≤ 4 for x ∈ [-1, 1].
A2.  R is the reals. For functions f, g : R → R and x ∈ R define I(fg) = ∫1x f(t) g(t) dt. If a(x), b(x), c(x), d(x) are real polynomials, show that I(ac) I(bd) - I(ad) I(bc) is divisible by (x - 1)4.
A3.  ABCD are the vertices of a square with A opposite C and side AB = s. The distances of a point P in space from A, B, C, D are a, b, c, d respectively. Show that a2 + c2 = b2 + d2, and that the perpendicular distance k of P from the plane ABCD is given by 8k2 = 2(a2 + b2 + c2 + d2) - 4s2 - (a4 + b4 + c4 + d4 - 2a2c2 - 2b2d2)/s2.
A4.  R is the reals. f : R → R has a continuous derivative, f(0) = 0, and |f '(x)| ≤ |f(x)| for all x. Show that f is constant.
A5.  Let T be a tangent plane to the ellipsoid x2/a2 + y2/b2 + z2/c2 = 1. What is the smallest possible volume for the tetrahedral volume bounded by T and the planes x = 0, y = 0, z = 0?
A6.  A particle moves in one dimension. Its distance x from the origin at time t is at + bt2 + ct3. Find an expression for the particle's acceleration in terms of a, b, c and its speed v.
B1.  Two circles C1 and C2 intersect at A and B. C1 has radius 1. L denotes the arc AB of C2 which lies inside C1. L divides C1 into two parts of equal area. Show L has length > 2.
B2.  P0 is the parabola y2 = mx, vertex K (0, 0). If A and B points on P0 whose tangents are at right angles, let C be the centroid of the triangle ABK. Show that the locus of C is a parabola P1. Repeat the process to define Pn. Find the equation of Pn.
B3.  The density of a solid sphere depends solely on the radial distance. The gravitational force at any point inside the sphere, a distance r from the center, is kr2 (where k is a constant). Find the density (in terms of G, k and r), and the gravitational force at a point outside the sphere. [You may assume the usual results about the gravitational attraction of a spherical shell.]
B4.  Define an = 2(1 + 1/n)2n+1/( (1 + 1/n)n + (1 + 1/n)n+1). Prove that an is strictly monotonic increasing.
B5.  Let m be the smallest integer greater than (√3 + 1)2n. Show that m is divisible by 2n+1.
B6.  The particle P moves in the plane. At t = 0 it starts from the point A with velocity zero. It is next at rest at t = T, when its position is the point B. Its path from A to B is the arc of a circle center O. Prove that its acceleration at each point in the time interval [0, T] is non-zero, and that at some point in the interval its acceleration is directly towards the center O.

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