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 P_k 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 lim_x→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 a_n satisfies a_n = ∑_n+1^∞ a_k². Show ∑ a_n does not converge unless all a_n 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 a_n = ∑₁ⁿ (-1)ⁱ⁺¹/i. Assume that lim_n→∞ a_n = 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 b_n be the sum of the first n terms of the rearranged series. Assume that lim_n→∞ b_n = h. Show that b_3n = a_4n + a_2n/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 a_n be a sequence in (-1, 0) tending to zero and b_n a sequence in (0, 1) tending to zero. Show that lim_n→∞ (f(b_n) - f(a_n)/(b_n - a_n) = f '(0).
B6. If x is a positive rational, show that we can find distinct positive integers a₁, a₂, ... , a_n such that x = ∑ 1/a_i.
B7. Let α be a positive real. Let a_n = S₁ⁿ (α/n + i/n)ⁿ. Show that lim a_n ∈ (e^α, e^α+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.