6th British Mathematical Olympiad 1970 Problems

6th British Mathematical Olympiad 1970 Problems

1.  (1) Find 1/log₂a + 1/log₃a + ... + 1/log_na 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) ∫ _-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.

7.  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.  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) = r^1/4 - s^1/4.

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?