### 49th Putnam 1988

 A1.  Let S = { (x, y) : |x| - |y| ≤ 1 and |y| ≤ 1 }. Sketch S and find its area. A2.  Let f(x) = exp(x2). Find an open interval I and a non-zero function g(x) on I such that (fg)' = f 'g' on I, or prove that they do not exist. A3.  For what real numbers α does (1/ 1 cosec(1) - 1)α + (1/2 cosec(1/2) - 1)α + ... + (1/n cosec(1/n) - 1)α + ... converge? A4.  The plane is divided into 3 disjoint sets. Can we always find two points in the same set a distance 1 apart? What about 9 disjoint sets? A5.  R+ denotes the positive reals. Prove that there is a unique function f : R+ → R+ satisfying f( f(x) ) = 6x - f(x) for all x. A6.  V is an n-dimensional vector space. Can we find a linear map A : V → V with n+1 eigenvectors, any n of which are linearly independent, which is not a scalar multiple of the identity? B1.  If n > 3 is not prime, show that we can find positive integers a, b, c, such that n = ab + bc + ca + 1. B2.  α is a non-negative real. x is a real satisfying (x + 1)2 ≥ α(α + 1). Is x2 ≥ α(α - 1)? B3.  αn is the smallest element of the set { |a - b√3| : a, b non-negative integers with sum n}. Find sup αn. B4.  αn are positive reals, and βn = αnn/(n+1). Show that if ∑ αn converges, then so does ∑ βn. B5.  Find the rank of the 2n+1 x 2n+1 skew-symmetric matrix with entries given by aij = 1 for (i - j) = -2n, -(2n-1), ... , -(n+1); -1 for (i - j) = -n, -(n-1), ... , -1; 1 for (i - j) = 1, 2, ... , n; -1 for (i - j) = n+1, n+2, ... , 2n+1. In other words, the main diagonal is 0s, the n diagonals immediately below the main diagonal are 1s, the n diagonals below that are -1s, the n diagonals immediately above the main diagonal are -1s, and the n diagonals above that are 1s. B6.  The triangular numbers are 1, 3, 6, 10, ... , n(n+1)/2, ... . Prove that there are an infinite number of pairs ai, bi such that m is triangular iff aim + bi is triangular.

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 96 (1989) 690-5.

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