The Sixth British Mathematical Olympiad
1970年第六届英国奥林匹克数学竞赛 
 (1) Find 1/log_{2}a + 1/log_{3}a + ... + 1/log_{n}a as a quotient of two logs to base 2.
(2) Find the sum of the coefficients of (1 + x  x^{2})^{3}(1  3x + x^{2})^{2} and the sum of the coefficients of its derivative.
 Sketch the curve x^{2} + 3xy + 2y^{2} + 6x + 12y + 4. Where is the center of symmetry ?
 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 ?
 Prove that any subset of a set of n positive integers has a nonempty subset whose sum is divisible by n.
 What is the minimum number of planes required to divide a cube into at least 300 pieces ?
 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) ∫ _{a}^{a} y(x) dx = 2a y(0). If you integrate numerically from (a, 0) using 2N equal steps δ using g(x_{n+1}) = g(x_{n}) + δ x g'(x_{n}), then the resulting solution does not satisfy (1). Suggest a modified method which ensures that (1) is satisfied .
 ABC is a triangle with ∠B = ∠C = 50^{o}. D is a point on BC and E a point on AC such that ∠BAD = 50^{o} and ∠ABE = 30^{o}. Find ∠BED.
 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 ?
 Find rationals r and s such that √(2√3  3) = r^{1/4}  s^{1/4} .
 Find "some kind of 'formula' for" the number f(n) of incongruent rightangled triangles with shortest side n ? Show that f(n) is unbounded. Does it tend to infinity ?

点此查看相关视频讲解 

