### 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-2 ∫D a(x, y, R) dx dy. A5.  Let α1, α2, ... , αn be the nth roots of unity. Find ∏i ∑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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