International Maths Olympiad 1999

Each day's Paper contains three questions, each question being worth 7 points.

Contestants are allowed four and a half hours for each day's paper.

 

FIRST DAY

1. Determine all finite sets S of at least three points in the plane which satisfy the following condition:

   for any two distinct points A and B in S, the perpendicular bisector of the line segment AB is an axis of symmetry for S.

    

2. Let n be a fixed integer, with n>2 or n=2.

   (a) Determine the least constant C such that the inequality

holds for all real numbers x₁, ... , x_n >0 or =0

(b) For this constant C , determine when equality holds.

 

1. Consider an n x n square board, where n is a fixed even positive integer. The board is divided into n² unit squares. We say that two squares are adjacent if they have a common side. N unit squares on the board are marked in such a way that every square (marked or unmarked) on the board is adjacent to at least one marked square. Determine the smallest possible value of N (in terms of n).

 

SECOND DAY

1. Determine all pairs (n , p) of positive integers such that p is prime, n < 2p or n = 2p, and

(p - 1)ⁿ + 1 is divisible by n^{p - 1}.

1. Two circles C₁ and C₂ are contained inside a circle C , and are tangent to C at the distinct points M and N, respectively. C₁ passes through the centre of C₂. The line passing through the points of intersection of C₁ and C₂ meet C at A and B. The lines MA and MB meet C1 at D and E respectively. Prove that DE is tangent to C₂.
2. Determine all functions f : R --> R such that

f (x - f (y)) = f ( f (y)) + x f (y) + f (x) - 1 for all Real numbers x , y.

Return to Contest List Return to Resources Return to Home Page