61st Putnam 2000

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 ?
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.
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?
A4.  Show that limk→∞ &inf;0k sin x sin x2 dx converges.
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.
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.
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.
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.
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).
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].
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.
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.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.

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© John Scholes
1 Jan 2001