7th British Mathematical Olympiad 1971 Problems

7th British Mathematical Olympiad 1971 Problems 

1.  Factorise (a + b)⁷ - a⁷ - b⁷. Show that 2n³ + 2n² + 2n + 1 is never a multiple of 3.

2.  Let a = 9⁹ , b = 9^a, c = 9^b. 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, ... , a_n and b₁, b₂, ... , b_n at A and B respectively. Show that tan²a₁ + tan²a₂ + ... + tan²a_n = tan²b₁ + tan²b₂ + ... + tan²b_n.

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) = ∫_c^x f(x, u) du. Show that I'(x) = f(x, x) + ∫_c^x ∂f/∂x du. (2) Find lim_θ→0 cot θ sin(t sin θ).

(3) Let G(t) = ∫₀^t 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.