10th British Mathematical Olympiad 1974 Problems

10th British Mathematical Olympiad 1974 Problems

1.  C is the curve y = 4x²/3 for x ≥ 0 and C' is the curve y = 3x²/8 for x ≥ 0. Find curve C" which lies between them such that for each point P on C" the area bounded by C, C" and a horizontal line through P equals the area bounded by C", C and a vertical line through P.

2.  S is the set of all 15 dominoes (m, n) with 1 ≤ m ≤ n ≤ 5. Each domino (m, n) may be reversed to (n, m). How many ways can S be partitioned into three sets of 5 dominoes, so that the dominoes in each set can be arranged in a closed chain: (a, b), (b, c), (c, d), (d, e), (e, a)?

3.  Show that there is no convex polyhedron with all faces hexagons.

4.  A is the 16 x 16 matrix (a_i,j). a_1,1 = a_2,2 = ... = a_16,16 = a_16,1 = a_16,2 = ... = a_16,15 = 1 and all other entries are 1/2. Find A^-1.

5.  In a standard pack of cards every card is different and there are 13 cards in each of 4 suits. If the cards are divided randomly between 4 players, so that each gets 13 cards, what is the probability that each player gets cards of only one suit?

6.  ABC is a triangle. P is equidistant from the lines CA and BC. The feet of the perpendiculars from P to CA and BC are at X and Y. The perpendicular from P to the line AB meets the line XY at Z. Show that the line CZ passes through the midpoint of AB.

7.  b and c are non-zero. x³ = bx + c has real roots α, β, γ. Find a condition which ensures that there are real p, q, r such that β = pα² + qα + r, γ = pβ² qβ+ r, α = pγ² + qγ + r.

8.  p is an odd prime. The product (x + 1)(x + 2) ... (x + p - 1) is expanded to give a_p-1x^p-1 + ... + a₁x + a₀. Show that a_p-1 = 1, a_p-2 = p(p-1)/2!, 2a_p-3 = p(p-1)(p-2)/3! + a_p-2(p-1)(p-2)/2!, ... , (p-2)a₁ = p + a_p-2(p-1) + a_p-3(p-2) + ... + 3a₂, (p-1)a₀ = 1 + a_p-2 + ... + a₁. Show that a₁, a₂, ... , a_p-2 are divisible by p and (a₀ + 1) is divisible by p. Show that for any integer x, (x+1)(x+2) ... (x+p-1) - x^p-1 + 1 is divisible by p. Deduce Wilson's theorem that p divides (p-1)! + 1 and Fermat's theorem that p divides x^p-1 - 1 for x not a multiple of p.

9.  A uniform rod is attached by a frictionless joint to a horizontal table. At time zero it is almost vertical and starts to fall. How long does it take to reach the table? You may assume that ∫ cosec x dx = log |tan x/2|.

10.  A long solid right circular cone has uniform density, semi-vertical angle x and vertex V. All points except those whose distance from V lie in the range a to b are removed. The resulting solid has mass M. Show that the gravitational attraction of the solid on a point of unit mass at V is 3/2 GM(1 + cos x)/(a² + ab + b²).