46th Putnam 1985

A1.  How many triples (A, B, C) are there of sets with A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A ∩ B ∩ C = ∅?
A2.  ABC is an acute-angled triangle with area 1. A rectangle R has its vertices Ri on the sides of the triangle, R1 and R2 on BC, R3 on AC and R4 on AB. Another rectangle S has it vertices on the sides of the triangle AR3R4, two on R3R4 and one one each of the other two sides. What is the maximum total area of R and S over all possible choices of triangle and rectangles?
A3.  x is a real. Define ai 0 = x/2i, ai j+1 = ai j2 + 2 ai j. What is limn→∞ an n?
A4.  Let an be the sequence defined by a1 = 3, an+1 = 3k, where k = an. Let bn be the remainder when an is divided by 100. Which values bn occur for infinitely many n?
A5.  Let fn(x) = cos x cos 2x ... cos nx. For which n in the range 1, 2, ... , 10 is ∫0 fn(x) dx non-zero?
A6.  Find a polynomial f(x) with real coefficients and f(0) = 1, such that the sums of the squares of the coefficients of f(x)n and (3x2 + 7x + 2)n are the same for all positive integers n.
B1.  p(x) is a polynomial of degree 5 with 5 distinct integral roots. What is the smallest number of non-zero coefficients it can have? Give a possible set of roots for a polynomial achieving this minimum.
B2.  The polynomials pn(x) are defined as follows: p0(x) = 1; pn+1'(x) = (n + 1) pn(x + 1), pn+1(0) = 0 for n ≥ 0. Factorize p100(1) into distinct primes.
B3.  ai j is a positive integer for i, j = 1, 2, 3, ... and for each positive integer we can find exactly eight ai j equal to it. Prove that ai j > ij for some i, j.
B4.  Let C be the circle radius 1, center the origin. A point P is chosen at random on the circumference of C, and another point Q is chosen at random in the interior of C. What is the probability that the rectangle with diagonal PQ, and sides parallel to the x-axis and y-axis, lies entirely inside (or on) C?
B5.  Assuming that ∫-∞ e-x2 dx = √π, find ∫0 x-1/2 e-1985(x + 1/x) dx.
B6.  G is a finite group consisting of real n x n matrices with the operation of matrix multiplication. The sum of the traces of the elements of G is zero. Prove that the sum of the elements of G is the zero matrix.

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 Society 93 (1986) 621-6.


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4 Jan 2001