38th Putnam 1977

A1.  Show that if four distinct points of the curve y = 2x4 + 7x3 + 3x - 5 are collinear, then their average x-coordinate is some constant k. Find k.
A2.  Find all real solutions (a, b, c, d) to the equations a + b + c = d, 1/a + 1/b + 1/c = 1/d.
A3.  R is the reals. f, g, h are functions R → R. f(x) = (h(x + 1) + h(x - 1) )/2, g(x) = (h(x + 4) + h(x - 4) )/2. Express h(x) in terms of f and g.
A4.  Find polynomials p(x) and q(x) with integer coefficients such that p(x)/q(x) = ∑0 x2n/(1 - x2n+1) for x ∈ (0, 1).
A5.  p is a prime and m ≥ n are non-negative integers. Show that (pm)C(pn) = mCn (mod p), where mCn is the binomial coefficient.
A6.  R is the reals. X is the square [0, 1] x [0, 1]. f : X → R is continuous. If ∫Y f(x, y) dx dy = 0 for all squares Y such that (1) Y ⊆ X, (2) Y has sides parallel to those of X, (3) at least one of Y's sides is contained in the boundary of X, is it true that f(x, y) = 0 for all x, y?
B1.  Find ∏2 (n3 - 1)/(n3 + 1).
B2.  P is a plane containing a convex quadrilateral ABCD. X is a point not in P. Find points A', B', C', D' on the lines XA, XB, XC, XD respectively so that A'B'C'D' is a parallelogram.
B3.  Let S be the set of all collections of 3 (not necessarily distinct) positive irrational numbers with sum 1. If A = {x, y, z} ∈ S and x > 1/2, define A' = {2x - 1, 2y, 2z}. Does repeated application of this operation necessarily give a collection with all elements < 1/2?
B4.  Let P be a point inside a continuous closed curve in the plane which does not intersect itself. Show that we can find two points on the curve whose midpoint is P.
B5.  a1, a2, ... , an are real and b < (∑ ai)2/(n - 1) - ∑ ai2. Show that b < 2aiaj for all distinct i, j.
B6.  G is a group. H is a finite subgroup with n elements. For some element g ∈ G, (gh)3 = 1 for all elements h ∈ H. Show that there are at most 3n2 distinct elements which can be written as a product of a finite number of elements of the coset Hg.

To avoid possible copyright problems, I have changed the wording, but not the substance, of all the problems. The original text and official solutions were published in American Mathematical Monthly 86 (1979) 170-5. They are also available (with the solutions expanded) in: Gerald L Alexanderson et al, The William Lowell Putnam Mathematical Competition, 1965-1984. Out of print, but in some university libraries.

Putnam home
© John Scholes
9 Oct 1999