### 41st Putnam 1980

 A1.  Let f(x) = x2 + bx + c. Let C be the curve y = f(x) and let Pi be the point (i, f(i) ) on C. Let Ai be the point of intersection of the tangents at Pi and Pi+1. Find the polynomial of smallest degree passing through A1, A2, ... , A9. A2.  Find f(m, n), the number of 4-tuples (a, b, c, d) of positive integers such that the lowest common multiple of any three integers in the 4-tuple is 3m7n. A3.  Find ∫0π/2 f(x) dx, where f(x) = 1/(1 + tan√2x). A4.  Show that for any integers a, b, c, not all zero, and such that |a|, |b|, |c| < 106, we have |a + b √2 + c √3| > 10-21. But show that we can find such a, b, c with |a + b √2 + c √3| < 10-11. A5.  Let p(x) be a polynomial with real coefficients of degree 1 or more. Show that there are only finitely many values α such that ∫0α p(x) sin x dx = 0 and ∫0α p(x) cos x dx = 0. A6.  Let R be the reals and C the set of all functions f : [0, 1] → R with a continuous derivative and satisfying f(0) = 0, f(1) = 1. Find infC ∫01 | f '(x) - f(x) | dx. B1.  For which real k do we have cosh x ≤ exp(k x2) for all real x? B2.  S is the region of space defined by x, y, z ≥ 0, x + y + z ≤ 11, 2x + 4y + 3z ≤ 36, 2x + 3z ≤ 24. Find the number of vertices and edges of S. For which a, b is ax + by + z ≤ 2a + 5b + 4 for all points of S? B3.  Define an by a0 = α, an+1 = 2an - n2. For which α are all an positive? B4.  S is a finite set with subsets S1, S2, ... , S1066 each containing more than half the elements of S. Show that we can find T ⊆ S with |T| ≤ 10, such that all T ∩ Si are non-empty. B5.  R0+ is the non-negative reals. For α ≥ 0, Cα is the set of continuous functions f : [0, 1] → R0+ such that: (1) f is convex [ f(λx + μy) ≤ λf(x) + μf(y) for λ, μ ≥ 0 with λ + μ = 1]; (2) f is increasing; (3) f(1) - 2 f(2/3) + f(1/3) ≥ α ( f(2/3) - 2 f(1/3) + f(0) ). For which α is Cα is closed under pointwise multiplication? B6.  An array of rationals f(n, i) where n and i are positive integers with i > n is defined by f(1, i) = 1/i, f(n+1, i) = (n+1)/i ( f(n, n) + f(n, n+1) + ... + f(n, i - 1) ). If p is prime, show that f(n, p) has denominator (when in lowest terms) not a multiple of p (for n > 1).

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 88 (1981) 607-12. 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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