27th Putnam 1966

A1.  Let f(n) = ∑1n [r/2]. Show that f(m + n) - f(m - n) = mn for m > n > 0.
A2.  A triangle has sides a, b, c. The radius of the inscribed circle is r and s = (a + b + c)/2. Show that 1/(s - a)2 + 1/(s - b)2 + 1/(s - c)2 ≥ 1/r2.
A3.  Define the sequence {an} by a1 ∈ (0, 1), and an+1 = an(1 - an). Show that limn→∞n an = 1.
A4.  Delete all the squares from the sequence 1, 2, 3, ... . Show that the nth number remaining is n + m, where m is the nearest integer to √n.
A5.  Let S be the set of continuous real-valued functions on the reals. φ :S → S is a linear map such that if f, g ∈ S and f(x) = g(x) on an open interval (a, b), then φf = φg on (a, b). Prove that for some h ∈ S, (φf)(x) = h(x)f(x) for all f and x.
A6.  Let an = √(1 + 2 √(1 + 3 √(1 + 4 √(1 + 5 √( ... + (n - 1) √(1 + n) ... ) ) ) ) ). Prove lim an = 3.
B1.  A convex polygon does not extend outside a square side 1. Prove that the sum of the squares of its sides is at most 4.
B2.  Prove that at least one integer in any set of ten consecutive integers is relatively prime to the others in the set.
B3.  an is a sequence of positive reals such that ∑ 1/an converges. Let sn = ∑1n ai. Prove that ∑ n2an/sn2 converges.
B4.  Given a set of (mn + 1) unequal positive integers, prove that we can either (1) find m + 1 integers biin the set such that bi does not divide bj for any unequal i, j, or (2) find n+1 integers ai in the set such that ai divides ai+1 for i = 1, 2, ... , n.
B5.  Given n points in the plane, no three collinear, prove that we can label them Pi so that P1P2P3 ... Pn is a simple closed polygon (with no edge intersecting any other edge except at its endpoints).
B6.  y = f(x) is a solution of y'' + exy = 0. Prove that f(x) is bounded.

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 74 (1967) 772-7. 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
9 Oct 1999