33rd Putnam 1972

A1.  Show that we cannot have 4 binomial coefficients nCm, nC(m+1), nC(m+2), nC(m+3) with n, m > 0 (and m + 3 ≤ n) in arithmetic progression.
A2.  Let S be a set with a binary operation * such that (1) a * (a * b) = b for all a, b ∈ S, (2) (a * b) * b = a for all a, b ∈ S. Show that * is commutative. Give an example for which S is not associative.
A3.  A sequence { xi } is said to have a Cesaro limit iff limn→∞(x1 + x2 + ... + xn)/n exists. Find all (real-valued) functions f on the closed interval [0, 1] such that { f(xi) } has a Cesaro limit iff { xi } has a Cesaro limit.
A4.  Show that a circle inscribed in a square has a larger perimeter than any other ellipse inscribed in the square.
A5.  Show that n does not divide 2n - 1 for n > 1.
A6.  f is an integrable real-valued function on the closed interval [0, 1] such that ∫01 xmf(x) dx = 0 for m = 0, 1, 2, ... , n - 1, and 1 for m = n. Show that |f(x)| ≥ 2n(n + 1) on a set of positive measure.
B1.  Let ∑0 xn(x - 1)2n / n! = ∑0 an xn. Show that no three consecutive an are zero.
B2.  A particle moves in a straight line with monotonically decreasing acceleration. It starts from rest and has velocity v a distance d from the start. What is the maximum time it could have taken to travel the distance d?
B3.  A group has elements g, h satisfying: ghg = hg2h, g3 = 1, hn = 1 for some odd n. Prove h = 1.
B4.  Show that for n > 1 we can find a polynomial p(a, b, c) with integer coefficients such that p(xn, xn+1, x + xn+2) ≡ x.
B5.  A, B, C and D are non-coplanar points. ∠ABC = ∠ADC and ∠BAD = ∠BCD. Show that AB = CD and BC = AD.
B6.  The polynomial p(x) has all coefficients 0 or 1, and p(0) = 1. Show that if the complex number z is a root, then |z| ≥ (√5 - 1)/2.


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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 80 (1973) 1019-28. 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.

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© John Scholes
22 Aug 2001