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 R_i 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 a_{i 0} = x/2ⁱ, a_{i j+1} = a_{i j}² + 2 a_{i j}. What is lim_n→∞ a_{n n}?
A4. Let a_n be the sequence defined by a₁ = 3, a_n+1 = 3^k, where k = a_n. Let b_n be the remainder when a_n is divided by 100. Which values b_n occur for infinitely many n?
A5. Let f_n(x) = cos x cos 2x ... cos nx. For which n in the range 1, 2, ... , 10 is ∫₀^2π f_n(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 p_n(x) are defined as follows: p₀(x) = 1; p_n+1'(x) = (n + 1) p_n(x + 1), p_n+1(0) = 0 for n ≥ 0. Factorize p₁₀₀(1) into distinct primes.
B3. a_{i j} is a positive integer for i, j = 1, 2, 3, ... and for each positive integer we can find exactly eight a_{i j} equal to it. Prove that a_{i 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.