51st Putnam 1990

A1.  Prove that the sequence a0 = 2, 3, 6, 14, 40, 152, 784, ... with general term an = (n+4) an-1 - 4n an-2 + (4n-8) an-3 is the sum of two well-known sequences.
A2.  Can we find a subsequence of { n1/3 - m1/3 : n, m = 0, 1, 2, ... } which converges to √2?
A3.  A convex pentagon has all its vertices lattice points in the plane (and no three collinear). Prove that its area is at least 5/2.
A4.  Given a point P in the plane, let SP be the set of points whose distance from P is irrational. What is the smallest number of such sets whose union is the entire plane?
A5.  M and N are n x n matrices such that (MN)2 = 0. Must (NM)2 = 0?
A6.  How many ordered pairs (A, B) of subsets of {1, 2, ... , 10} can we find such that each element of A is larger than |B| and each element of B is larger than |A|.
B1.  R is the real line. Find all possible functions f: R → R with continuous derivative such that f(α)2 = 1990 + ∫0α ( f(x)2 + f '(x)2) dx for all α.
B2.  Let Pn(x, z) = ∏1n (1 - z xi-1) / (z - xi). Prove that 1 + ∑1 (1 + xn) Pn(x, z) = 0 for |z| > 1 and |x| < 1.
B3.  Let S be the set of 2 x 2 matrices each of whose elements is one of the 15 squares 0, 1, 4, ... , 196. Prove that if we select more than 154 - 152 - 15 + 2 matrices from S, then two of those selected must commute.
B4.  A finite group with n elements is generated by g and h. Can we arrange two copies of the elements of the group in a sequence (total length 2n) so that each element is g or h times the previous element and the first element is g or h times the last?
B5.  Can we find a sequence of reals αi ≠ 0 such that each polynomial α0 + α1x + ... + αnxn has all its roots real and distinct?
B6.  C is a non-empty, closed, bounded, convex subset of the plane. Given a support line L of C and a real number 0 ≤ α ≤ 1, let Bα be the band parallel to L, situated midway between L and the parallel support line on the other side of C, and of width α times the distance between the two support lines. What is the smallest α such that ∩ Bα contains a point of C, where the intersection is taken over all possible directions for the support line L?

To avoid possible copyright problems, I have changed the wording, but not the substance of all the problems. The original text of the problems and the official solutions are in American Mathematical Monthly 98 (1991) 721-7.  
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© John Scholes
16 Dec 1998