8th Putnam 1948

A1.  C is the complex numbers. f : C → R is defined by f(z) = |z3 - z + 2|. What is the maximum value of f on the unit circle |z| = 1?
A2.  K is a cone. s is a sphere radius r, and S is a sphere radius R. s is inside K touches it along all points of a circle. S is also inside K and touches it along all points of a circle. s and S also touch each other. What is the volume of the finite region between the two spheres and inside K?
A3.  an is a sequence of positive reals decreasing monotonically to zero. bn is defined by bn = an - 2an+1 + an+2 and all bn are non-negative. Prove that b1 + 2b2 + 3b3 + ... = a1.
A4.  Let D be a disk radius r. Given (x, y) ∈ D, and R > 0, let a(x, y, R) be the length of the arc of the circle center (x, y), radius R, which is outside D. Evaluate limR→0 R-2D a(x, y, R) dx dy.
A5.  Let α1, α2, ... , αn be the nth roots of unity. Find ∏i<ji - αj)2.
A6.  Do either (1) or (2):
(1) On each element ds of a closed plane curve there is a force 1/R ds, where R is the radius of curvature. The force is towards the center of curvature at each point. Show that the curve is in equilibrium.
(2) Prove that x + 2/3 x3 + 2.4/3.5 x5 + ... + 2.4. ... .2n/(3.5. ... .2n+1) x2n+1 + ... = (1 - x2)-1/2 sin-1x
B1.  p(x) is a cubic polynomial with roots α, β, γ and p'(x) divides p(2x). Find the ratios α : β : γ.
B2.  A circle radius r is tangent to the three coordinate planes (x =0, y =0, z = 0) in space. Find the locus of its center.
B3.  Show that [√n + √(n + 1)] = [√(4n + 2)] for positive integers n.
B4.  R is the reals. For what λ can we find a continuous function f : (0, 1) → R, not identically zero, such that ∫01 min(x, y) f(y) dy = λ f(x) for all x ∈ (0, 1)?
B5.  Find the area of the region { (x, y) : |x + yt + t2| ≤ 1 for all t ∈ [0, 1] }.
B6.  Do either (1) or (2):
(1) Take the origin O of the complex plane to be the vertex of a cube, so that OA, OB, OC are edges of the cube. Let the feet of the perpendiculars from A, B, C to the complex plane be the complex numbers u, v, w. Show that u2 + v2 + w2 = 0.
(2) Let (aij) be an n x n matrix. Suppose that for each i, 2 |aii| > ∑1n |aij|. By considering the corresponding system of linear equations or otherwise, show that det aij ≠ 0.

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