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The Seventh British Mathematical Olympiad
1971年第七届英国奥林匹克数学竞赛
  1. Factorise (a + b)7 - a7 - b7. Show that 2n3 + 2n2 + 2n + 1 is never a multiple of 3.
  2. Let a = 99 , b = 9a, c = 9b. Show that the last two digits of b and c are equal. What are they ?
  3. A and B are two vertices of a regular 2n-gon. The n longest diameters subtend angles a1, a2, ... , an and b1, b2, ... , bn at A and B respectively. Show that tan2a1 + tan2a2 + ... + tan2an = tan2b1 + tan2b2 + ... + tan2bn .
  4. Given any n+1 distinct integers all less than 2n+1, show that there must be one which divides another .
  5. The triangle ABC has circumradius R. ∠A ≥ ∠B ≥ ∠C. What is the upper limit for the radius of circles which intersect all three sides of the triangle ?
  6. (1) Let I(x) = ∫cx f(x, u) du. Show that I'(x) = f(x, x) + ∫cx ∂f/∂x du.(此处可能不能正常显示。请调整浏览器设置。)
    (2) Find limθ→0 cot θ sin(t sin θ).
    (3) Let G(t) = ∫0t cot θ sin(t sin θ) dθ. Prove that G'(π/2) = 2/π.
  7. Find the probability that two points chosen at random on a segment of length h are a distance less than k apart .
  8. A is a 3 x 2 real matrix, B is a 2 x 3 real matrix. AB = M where det M = 0 BA = det N where det N is non-zero, and M2 = kM. Find det N in terms of k .
  9. A solid spheres is fixed to a table. Another sphere of equal radius is placed on top of it at rest. The top sphere rolls off. Show that slipping occurs then the line of centers makes an angle θ to the vertical, where 2 sin θ = μ(17 cos θ - 10). Assume that the top sphere has moment of inertia 2/5 Mr2 about a diameter, where r is its radius .
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