### 39th Putnam 1978

 A1.  Let S = {1, 4, 7, 10, 13, 16, ... , 100}. Let T be a subset of 20 elements of S. Show that we can find two distinct elements of T with sum 104. A2.  Let A be the real n x n matrix (aij) where aij = a for i < j, b (≠ a) for i > j, and ci for i = j. Show that det A = (b p(a) - a p(b) )/(b - a), where p(x) = ∏ (ci - x). A3.  Let p(x) = 2(x6 + 1) + 4(x5 + x) + 3(x4 + x2) + 5x3. Let a = ∫0∞ x/p(x) dx, b = ∫0∞ x2/p(x) dx, c = ∫0∞ x3/p(x) dx, d = ∫0∞ x4/p(x) dx. Which of a, b, c, d is the smallest? A4.  A binary operation (represented by multiplication) on S has the property that (ab)(cd) = ad for all a, b, c, d. Show that: (1) if ab = c, then cc = c; (2) if ab = c, then ad = cd for all d. Find a set S, and such a binary operation, which also satisfies: (A) a a = a for all a; (B) ab = a ≠ b for some a, b; (C) ab ≠ a for some a, b. A5.  Let a1, a2, ... , an be reals in the interval (0, π) with arithmetic mean μ. Show that ∏ (sin ai)/ai ≤ ( (sin μ)/μ )n. A6.  Given n points in the plane, prove that less than 2n3/2 pairs of points are a distance 1 apart. B1.  A convex octagon inscribed in a circle has 4 consecutive sides length 3 and the remaining sides length 2. Find its area. B2.  Find ∑1∞∑1∞ 1/(i2j + 2ij + ij2). B3.  The polynomials pn(x) are defined by p1(x) = 1 + x, p2(x) = 1 + 2x, p2n+1(x) = p2n(x) + (n + 1) x p2n-1(x), p2n+2(x) = p2n+1(x) + (n + 1) x p2n(x). Let an be the largest real root of pn(x). Prove that an is monotonic increasing and tends to zero. B4.  Show that we can find integers a, b, c, d such that a2 + b2 + c2 + d2 = abc + abd + acd + bcd, and the smallest of a, b, c, d is arbitarily large. B5.  Find the real polynomial p(x) of degree 4 with largest possible coefficient of x4 such that p( [-1, 1] ) ⊆ [0, 1]. B6.  aij are reals in [0, 1]. Show that ( ∑i=1n ∑j=1mi aij/i )2 ≤ 2m ∑i=1n ∑j=1mi aij.

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) 751-7. 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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