| A1. Show that we cannot have 4 binomial coefficients nCm, nC(m+1), nC(m+2), nC(m+3) with n, m > 0 (and m + 3 ≤ n) in arithmetic progression. |
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| A2. Let S be a set with a binary operation * such that (1) a * (a * b) = b for all a, b ∈ S, (2) (a * b) * b = a for all a, b ∈ S. Show that * is commutative. Give an example for which S is not associative. |
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| A3. A sequence { xi } is said to have a Cesaro limit iff limn→∞(x1 + x2 + ... + xn)/n exists. Find all (real-valued) functions f on the closed interval [0, 1] such that { f(xi) } has a Cesaro limit iff { xi } has a Cesaro limit. |
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| A4. Show that a circle inscribed in a square has a larger perimeter than any other ellipse inscribed in the square. |
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| A5. Show that n does not divide 2n - 1 for n > 1. |
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| A6. f is an integrable real-valued function on the closed interval [0, 1] such that ∫01 xmf(x) dx = 0 for m = 0, 1, 2, ... , n - 1, and 1 for m = n. Show that |f(x)| ≥ 2n(n + 1) on a set of positive measure. |
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| B1. Let ∑0∞ xn(x - 1)2n / n! = ∑0∞ an xn. Show that no three consecutive an are zero. |
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| B2. A particle moves in a straight line with monotonically decreasing acceleration. It starts from rest and has velocity v a distance d from the start. What is the maximum time it could have taken to travel the distance d? |
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| B3. A group has elements g, h satisfying: ghg = hg2h, g3 = 1, hn = 1 for some odd n. Prove h = 1. |
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| B4. Show that for n > 1 we can find a polynomial p(a, b, c) with integer coefficients such that p(xn, xn+1, x + xn+2) ≡ x. |
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| B5. A, B, C and D are non-coplanar points. ∠ABC = ∠ADC and ∠BAD = ∠BCD. Show that AB = CD and BC = AD. |
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| B6. The polynomial p(x) has all coefficients 0 or 1, and p(0) = 1. Show that if the complex number z is a root, then |z| ≥ (√5 - 1)/2. |
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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 80 (1973) 1019-28. 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
22 Aug 2001