18th British Mathematical Olympiad 1982 Problems

18th BMO 1982 - Further International Selection Test

1.  ABC is a triangle. The angle bisectors at A, B, C meet the circumcircle again at P, Q , R respectively. Show that AP + BQ + CR > AB + BC + CA.

2.  The sequence p₁, p₂, p₃, ... is defined as follows. p₁ = 2. p_n+1 is the largest prime divisor of p₁p₂ ... p_n + 1. Show that 5 does not occur in the sequence.

3.  a is a fixed odd positive integer. Find the largest positive integer n for which there are no positive integers x, y, z such that ax + (a + 1)y + (a + 2)z = n.

4.  a and b are positive reals and n > 1 is an integer. P₁ (x₁, y₁) and P₂ (x₂, y₂) are two points on the curve xⁿ - ayⁿ = b with positive real coordinates. If y₁ < y₂ and A is the area of the triangle OP₁P₂, show that   by₂ > 2ny₁^n-1a^1-1/nA.

5.  p(x) is a real polynomial such that p(2x) = 2^k-1(p(x) + p(x + 1/2) ), where k is a non-negative integer. Show that p(3x) = 3^k-1(p(x) + p(x + 1/3) + p(x + 2/3) ).aaa