2006.pdf

The 67th William Lowell Putnam Mathematical Competition consisted of 6 problems in two sections, A and B. The problems covered various mathematical topics, including geometry, number theory, algebra, and probability.

Problem A1 asked to find the volume of a region defined by an inequality involving x, y, and z. The solution involved changing to cylindrical coordinates and using Pappus's theorem to find the volume.

Problem A2 was a game theory problem where two players take turns removing stones from a heap. The goal was to prove that there are infinitely many values of n such that Bob has a winning strategy.

Problem A3 defined a sequence and asked to show that it has 2005 consecutive terms divisible by 2006. The solution involved rewriting the recursion and using the fact that the sequence is eventually periodic modulo any N.

Problem A4 asked to find the average number of local maxima of a permutation of a set S. The solution used the linearity of expectation and calculated the probability of having a local maximum at each position.

Problem A5 involved a polynomial with roots a1, ..., an and asked to find the value of a symmetric expression involving these roots. The solution used properties of complex numbers and polynomials to find the desired ratio.

Problem A6 asked to find the probability that four randomly chosen points in a circle form the vertices of a convex quadrilateral. The solution involved calculating the expected value of a random variable and using geometric probability.

The B problems covered topics such as geometry, linear algebra, and real analysis. Problem B1 asked to find the area of an equilateral triangle inscribed in a curve. Problem B2 involved finding a non-empty subset of a set of real numbers that satisfies a certain inequality. Problem B3 asked to find the maximum number of linear partitions of a set of points in the plane. Problem B4 involved finding the maximum number of points in a vector subspace of Rn that have coordinates 0 or 1. Problem B5 asked to find the maximum value of a functional involving a continuous function. Problem B6 involved evaluating a limit of a sequence defined recursively.