7th Balkan 1990 Problem 1 The sequence u n is defined by u 1 = 1, u 2 = 3, u n = (n+1) u n-1 - n u n-2 . Which members of the sequence which are divisible by 11? Solution
: all n except 1, 2, 3, 5, 6, 7, 9. We calculate the residues mod 11 to be: u 1 = 1, u 2 = 3, u 3 = -2, u 4 = 0, u 5 = -1, u 6 = 4, u 7 = 6, u 8 = 0, u 9 = 1, u 10 = 0, u 11 = 0. But now u n = 0 mod 11 for all n ≥ 11. So u n is divisible by 11 for n = 4, 8 and n ≥ 10. 7th Balkan 1990 © John Scholes jscholes@kalva.demon.co.uk 11 Apr 2002