4th Centromerican 2002 Problem A1 For which n > 2 can the numbers 1, 2, ... , n be arranged in a circle so that each number divides the sum of the next two numbers (in a clockwise direction)?
n=3 Solution Let the numbers be a 1 , a 2 , a 3 , ... . Where necessary we use cyclic subscripts (so that a n+1 means a 1 etc). Suppose a i and a i+1 are both even, then since a i divides a i+1 + a i+2 , a i+2 must also be even. Hence a i+3 must be even and so on. Contradiction, since only half the numbers are even. Hence if a i is even, a i+1 must be odd. But a i+1 + a i+2 must be even, so a i+2 must also be odd. In other words, every even number is followed by two odd numbers. But that means there are at least twice as many odd numbers as even numbers. That is only possible for n = 3. It is easy to check that n = 3 works. 4th OMCC 2002 © John Scholes jscholes@kalva.demon.co.uk 30 January 2004 Last corrected/updated 30 Jan 04