53rd Putnam 1992

A1.  Let Z be the integers. Prove that if f : Z → Z satisifies f( f(n) ) = f( f(n+2) + 2 ) = n for all n, and f(0) = 1, then f(n) = 1 - n.
A2.  Let the coefficient of x1992 in the power series (1 + x)α = 1 + αx + ... be C(α). Find ∫01 C(-y-1) ∑k=119921/(y+k) dy.
A3.  Find all positive integers a, b, m, n with m relatively prime to n such that (a2 + b2)m = (ab)n.
A4.  Let R be the reals. Let f : R → R be an infinitely differentiable function such that f(1/n) = n2/(n2+1) for n = 1, 2, 3, ... Find the value of the derivatives of f at zero: f(k)(0) for k = 1, 2, 3, ... .
A5.  Let N be the positive integers. Define f : N → {0, 1} by f(n) = 1 if the number of 1s in the binary representation of n is odd and 0 otherwise. Show that there do not exist positive integers k and m such that f(k + j) = f(k + m + j) = f(k + 2m + j) for 0 ≤ j < m.
A6.  Four points are chosen independently and at random on the surface of a sphere (using the uniform distribution). What is the probability that the center of the sphere lies inside the resulting tetrahedron?
B1.  Let R be the reals. Let S ⊆ R have n ≥ 2 elements. Let AS = { x ∈ R : x = (s + t)/2 for some s, t ∈ S with s ≠t}. What is the smallest possible |AS|?
B2.  Show that the coefficient of xk in the expansion of (1 + x + x2 + x3)n is ∑j=0k  nCj   nC(k-2j).
B3.  Let S be the set of points (x, y) in the plane such that the sequence an defined by a0 = x, an+1 = (an2 + y2)/2 converges. What is the area of S?
B4.  p(x) is a polynomial of degree < 1992 such that p(0), p(1), p(-1) are all non-zero. The 1992th derivative of p(x)/(x3 - x) = f(x)/g(x) for polynomials f(x) and g(x). Find the smallest possible degree of f(x).
B5.  Let An denote the n-1 x n-1 matrix (aij) with aij = i + 2 for i = j, and 1 otherwise. Is the sequence (det An)/n! bounded?
B6.  Let M be a set of real n x n matrices such that: (1) 1 ∈ M; (2) if A, B ∈ M, then just one of AB, - AB is in M; (3) if A, B ∈ M, then either AB = BA or AB = -BA; (4) if 1 ≠ A ∈ M, then there is at least one B ∈ M such that AB = - BA. Prove that M contains at most n2 matrices.

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 100 (1993) 757.  
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© John Scholes
16 Dec 1998