37th Putnam 1976

A1.  Given two rays OA and OB and a point P between them. Which point X on the ray OA has the property that if XP is extended to meet the ray OB at Y, then XP.PY has the smallest possible value.
A2.  Let a(x, y) be the polynomial x2y + xy2, and b(x, y) the polynomial x2 + xy + y2. Prove that we can find a polynomial pn(a, b) which is identically equal to (x + y)n + (-1)n (xn + yn). For example, p4(a, b) = 2b2.
A3.  Find all solutions to pn = qm ±1, where p and q are primes and m, n ≥ 2.
A4.  Let p(x) ≡ x3 + ax2 + bx - 1, and q(x) ≡ x3 + cx2 + dx + 1 be polynomials with integer coefficients. Let α be a root of p(x) = 0. p(x) is irreducible over the rationals. α + 1 is a root of q(x) = 0. Find an expression for another root of p(x) = 0 in terms of α, but not involving a, b, c, or d.
A5.  Let P be a convex polygon. Let Q be the interior of P and S = P ∪ Q. Let p be the perimeter of P and A its area. Given any point (x, y) let d(x, y) be the distance from (x, y) to the nearest point of S. Find constants α, β, γ such that ∫U e-d(x,y) dx dy = α + βp + γA, where U is the whole plane.
A6.  Let R be the real line. f : R → [-1, 1] is twice differentiable and f(0)2 + f '(0)2 = 4. Show that f(x0) + f ''(x0) = 0 for some x0.
B1.  Show that limn→∞ 1/n ∑1n ( [2n/i] - 2[n/i] ) = ln a - b for some positive integers a and b.
B2.  G is a group generated by the two elements g, h, which satisfy g4 = 1, g2 ≠ 1, h7 = 1, h ≠ 1, ghg-1h = 1. The only subgroup containing g and h is G itself. Write down all elements of G which are squares.
B3.  Let 0 < α < 1/4. Define the sequence pn by p0 = 1, p1 = 1 - α, pn+1 = pn - α pn-1. Show that if each of the events A1, A2, ... , An has probability at least 1 - α, and Ai and Aj are independent for | i - j | > 1, then the probability of all Ai occurring is at least pn. You may assume that all pn are positive.
B4.  Let an ellipse have center O and foci A and B. For a point P on the ellipse let d be the distance from O to the tangent at P. Show that PA·PB·d2 is independent of the position of P.
B5.  Find ∑0n (-1)i nCi ( x - i )n, where nCi is the binomial coefficient.
B6.  Let σ(n) be the sum of all positive divisors of n, including 1 and n. Show that if σ(n) = 2n + 1, then n is the square of an odd integer.

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 85 (1978) 28-33. 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