| A1. k is a positive constant. The sequence xi of positive reals has sum k. What are the possible values for the sum of xi2 ? |
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| A2. Show that we can find infinitely many triples N, N + 1, N + 2 such that each member of the triple is a sum of one or two squares. |
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| A3. An octagon is incribed in a circle. One set of alternate vertices forms a square area 5. The other set forms a rectangle area 4. What is the maximum possible area for the octagon? |
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| A4. Show that limk→∞ &inf;0k sin x sin x2 dx converges. |
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| A5. A, B, C each have integral coordinates and lie on a circle radius R. Show that at least one of the distances AB, BC, CA exceeds R1/3. |
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| A6. p(x) is a polynomial with integer coefficients. A sequence x0, x1, x2, ... is defined by x0 = 0, xn+1 = p(xn). Prove that if xn = 0 for some n > 0, then x1 = 0 or x2 = 0. |
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| B1. We are given N triples of integers (a1, b1, c1), (a2, b2, c2), ... (aN, bN, cN). At least one member of each triple is odd. Show that we can find integers A, B, C such that at least 4N/7 of the N values A ai + B bi + C ci are odd. |
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| B2. m and n are positive integers with m ≤ n. d is their greatest common divisor. nCm is the binomial coefficient. Show that d/n nCm is integral. |
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| B3. a1, a2, ... , aN are real and aN is non-zero. f(x) = a1 sin 2πx + a2 sin 4πx + a3 sin 6πx + ... + aN sin 2Nπx. Show that the number of zeros of f(i)(x) = 0 in the interval [0, 1) is a non-decreasing function of i and tends to 2N (as i tends to infinity). |
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| B4. f(x) is a continuous real function satisfying f(2x2 - 1) = 2 x f(x). Show that f(x) is zero on the interval [-1, 1]. |
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| B5. S0 is an arbitrary finite set of positive integers. Define Sn+1 as the set of integers k such that just one of k - 1, k is in Sn. Show that for infinitely many n, Sn is the union of S0 and a translate of S0. |
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| B6. Let X be the set of 2n points (±1, ±1, ... , ±1) in Euclidean n-space. Show that any subset of X with at least 2n+1/n points contains an equilateral triangle. |
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To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
Putnam home
© John Scholes
jscholes@kalva.demon.co.uk
1 Jan 2001