47th Putnam 1986

A1.  S is the set {x real st x4 - 13x2 + 36 ≤ 0}. Find the maximum value of f(x) = x3 - 3x on S.
A2.  What is the remainder when the integral part of 1020000/(10100 + 3) is divided by 10?
A3.  Find cot-1(1) + cot-1(3) + ... + cot-1(n2+n+1) + ... , where the cot-1(m) is taken to be the value in the range (0, π/2].
A4.  Let r(n) be the number of n x n matrices A = (aij) such that: (1) each aij = -1, 0, or 1; and (2) if we take any n elements aij, no two in the same row or column, then their sum is the same. Find rational numbers a, b, c, d, u, v, w such that r(n) = a un + b vn + c wn + d.
A5.  f:Rn→Rn is defined by f(x) =(f1(x), f2(x), ... , fn(x)), where x = (x1, x2, ... , xn) and the n functions fi:Rn→R have continuous 2nd order partial derivatives and satisfy ∂fi/∂xj - ∂fj/∂xi = cij (for all 1 ≤ i, j ≤ n) for some constants cij. Prove that there is a function g:Rn→R such that fi + ∂g/∂xi is linear (for all 1 ≤ i ≤ n).
A6.  Let p(x) be the real polynomial 1 + α1xm1 + α2xm2 + ... + αnxmn, where 0 < m1 < m2 < ... < mn. For some real polynomial q(x) we have p(x) = (1 - x)nq(x). Find q(1) in terms of m1, ... , mn.
B1.  ABCD is a rectangle. AEB is isosceles with E on the opposite side of AB to C and D and lies on the circle through A, B, C, D. This circle has radius 1. For what values of |AD| do the rectangle and triangle have the same area?
B2.  x, y, z are complex numbers satisfying x(x - 1) + 2yz = y(y - 1) + 2zx = z(z - 1) + 2xy. Prove that there are only finitely many possible values for the triple (x - y, y - z, z - x) and enumerate them.
B3.  We use the congruence notation for polynomials (in one variable) with integer coefficients to mean that the corresponding coefficients are congruent. Thus if f(x) = akxk + ... + a0, and g(x) = bkxk + ... + b0, then f = g (mod m) means that ai = bi (mod m) for all i. Let p be a prime and f(x), g(x), h(x), r(x), s(x) be polynomials with integer coefficients such that r(x)f(x) + s(x)g(x) = 1 (mod p) and f(x)g(x) = h(x) (mod p). Prove that for any positive integer n we can find F(x) and G(x) with integer coefficients such that F(x) = f(x) (mod p), G(x) = g(x) (mod p) and F(x)G(x) = h(x) (mod pn).
B4.  For real r > 0, define m(r) =min{ |r - √(m2+2n2)| for m, n integers}. Prove or disprove: (1) limr→∞ m(r) exists; and (2) is zero.
B5.  Let f(x, y, z) = x2 + y2 + z2 + xyz. a(x, y, z), b(x, y, z), b(x, y, z) are polynomials with real coefficients such that f(a(x, y, z), b(x, y, z), c(x, y, z)) = f(x, y, z). Prove or disprove that a(x, y, z), b(x, y, z), c(x, y, z) must be ±x, ±y, ±z in some order (with an even number of minus signs).
B6.  A, B, C, D are n x n matrices with entries in some field F. The transpose of a matrix X is denoted as X' (defined as X'ij = Xji). Given that A B' and C D' are symmetric, and A D' - B C' = 1, prove that A'D - C'B = 1.

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 94 (1987) 749-756.  
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© John Scholes
16 Dec 1998