25th British Mathematical Olympiad 1989 Problems

25th British Mathematical Olympiad 1989 Problems

1.  Find the smallest positive integer a such that ax² - bx + c = 0 has two distinct roots in the interval 0 < x < 1 for some integers b, c.

2.  Find the number of different ways of arranging five As, five Bs and five Cs in a row so that each letter is adjacent to an identical letter. Generalise to n letters each appearing five times.

3.  f(x) is a polynomial of degree n such that f(0) = 0, f(1) = 1/2, f(2) = 2/3, f(3) = 3/4, ... , f(n) = n/(n+1). Find f(n+1).

4.  D is a point on the side AC of the triangle ABC such that the incircles of BAD and BCD have equal radii. Express | BD | in terms of the lengths a = | BC |, b = | CA |, c = | AB |.

Problem 3

f(x) is a polynomial of degree n such that f(0) = 0, f(1) = 1/2, f(2) = 2/3, f(3) = 3/4, ... , f(n) = n/(n+1). Find f(n+1).

Solution

Put g(x) = (x+1)f(x)-x. Then g(0) = g(1) = ... = g(n) = 0, so g(x) = Ax(x-1)(x-2)...(x-n). g(-1) = 1, so A = (-1)ⁿ⁺¹/n+1!. Hence g(n+1) = (-1)ⁿ⁺¹. So if n is odd, g(n+1) = 1 and f(n+1) = 1. If n is even, then g(n+1) = -1 and f(n+1) = n/(n+2).

Thanks to Suat Namli