| 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. |
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| 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. |
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| A3. Find all solutions to pn = qm ±1, where p and q are primes and m, n ≥ 2. |
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| 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. |
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| 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. |
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| 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. |
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| B1. Show that limn→∞ 1/n ∑1n ( [2n/i] - 2[n/i] ) = ln a - b for some positive integers a and b. |
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| 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. |
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| 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. |
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| 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. |
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| B5. Find ∑0n (-1)i nCi ( x - i )n, where nCi is the binomial coefficient. |
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| 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. |
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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
jscholes@kalva.demon.co.uk
9 Oct 1999