14th Putnam 1954

A1.  Let N be the set {1, 2, ... , n}, where n is an odd integer. Let f : N x N → N satisfy: (1) f(r, s) = f(s, r) for all r, s; (2) {f(r, s) : s ∈ N} = N for each r. Show that {f(r, r) : r ∈ N} = N.
A2.  Given any five points in the interior of a square side 1, show that two of the points are a distance apart less than k = 1/√2. Is this result true for a smaller k?
A3.  Let S be the set of all curves satisfying y' + a(x) y = b(x), where a(x) and b(x) are never zero. Show that if C ∈ S, then the tangent at the point x = k on C passes through a point Pk which is independent of C.
A4.  A uniform rod length 2a is suspended in midair with one end resting against a smooth vertical wall at X and the other end attached by a string length 2b to a point on the wall above X. For what angles between the rod and the string is equilibrium possible?
A5.  R is the reals. f : (0, 1) → R satisfies limx→0 f(x) = 0, and f(x) - f(x/2) = o(x) as x→0. Show that f(x) = o(x) as x→0.
A6.  The real sequence an satisfies an = ∑n+1 ak2. Show ∑ an does not converge unless all an are zero.
A7.  Prove that the equation m2 + 3mn - 2n2 = 122 has no integral solutions.
B1.  Show that for any positive integer r, we can find integers m, n such that m2 - n2 = r3.
B2.  Let an = ∑1n (-1)i+1/i. Assume that limn→∞ an = k. Rearrange the terms by taking two positive terms, then one negative term, then another two positive terms, then another negative term and so on. Let bn be the sum of the first n terms of the rearranged series. Assume that limn→∞ bn = h. Show that b3n = a4n + a2n/2, and hence that h ≠ k.
B3.  Let S be a finite collection of closed intervals on the real line such that any two have a point in common. Prove that the intersection of all the intervals is non-empty.
B4.  Let F be a point, and L and D lines, in the plane. Show how to construct the point of intersection (if any) between L and the parabola with focus F and directrix D.
B5.  Let R be the reals. Let f : (-1, 1) → R be a function with a derivative at 0. Let an be a sequence in (-1, 0) tending to zero and bn a sequence in (0, 1) tending to zero. Show that limn→∞ (f(bn) - f(an)/(bn - an) = f '(0).
B6.  If x is a positive rational, show that we can find distinct positive integers a1, a2, ... , an such that x = ∑ 1/ai.
B7.  Let α be a positive real. Let an = S1n (α/n + i/n)n. Show that lim an ∈ (eα, eα+1).

To avoid possible copyright problems, I have changed the wording, but not the substance, of all the problems. The original text and solutions are available in: A M Gleason, R E Greenwood & L M Kelly, The William Lowell Putnam Mathematical Competition, Problems and Solutions, 1938-1964, MAA 1980. Out of print, but available in some university libraries.

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© John Scholes
27 Oct 1999
Last corrected/updated 30 Nov 03