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The Sixth British Mathematical Olympiad
1970年第六届英国奥林匹克数学竞赛
  1. (1) Find 1/log2a + 1/log3a + ... + 1/logna as a quotient of two logs to base 2.
    (2) Find the sum of the coefficients of (1 + x - x2)3(1 - 3x + x2)2 and the sum of the coefficients of its derivative.
  2. Sketch the curve x2 + 3xy + 2y2 + 6x + 12y + 4. Where is the center of symmetry ?
  3. Morley's theorem is as follows. ABC is a triangle. C' is the point of intersection of the trisector of angle A closer to AB and the trisector of angle B closer to AB. A' and B' are defined similarly. Then A'B'C' is equilateral. What is the largest possible value of area A'B'C'/area ABC? Is there a minimum value ?
  4. Prove that any subset of a set of n positive integers has a non-empty subset whose sum is divisible by n.
  5. What is the minimum number of planes required to divide a cube into at least 300 pieces ?
  6. y(x) is defined by y' = f(x) in the region |x| ≤ a, where f is an even, continuous function. Show that (1) y(-a) +y(a) = 2 y(0) and (2) ∫ -aa y(x) dx = 2a y(0). If you integrate numerically from (-a, 0) using 2N equal steps δ using g(xn+1) = g(xn) + δ x g'(xn), then the resulting solution does not satisfy (1). Suggest a modified method which ensures that (1) is satisfied .
  7. ABC is a triangle with ∠B = ∠C = 50o. D is a point on BC and E a point on AC such that ∠BAD = 50o and ∠ABE = 30o. Find ∠BED.
  8. 8 light bulbs can each be switched on or off by its own switch. State the total number of possible states for the 8 bulbs. What is the smallest number of switch changes required to cycle through all the states and return to the initial state ?
  9. Find rationals r and s such that √(2√3 - 3) = r1/4 - s1/4 .
  10. Find "some kind of 'formula' for" the number f(n) of incongruent right-angled triangles with shortest side n ? Show that f(n) is unbounded. Does it tend to infinity ?
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