42nd Swedish 2002 Problem 1 268 numbers are written around a circle. The 17th number is 3, the 83rd is 4 and the 144th is 9, The sum of every 20 consecutive numbers is 72. Find the 210th number.
-8/5 Solution Call the numbers a 1 , a 2 , ... , a 268 . We use the cyclic subscript convention, so that a 269 means a 1 etc. We have a n + a n+1 + ... + a n+19 = a n+1 + a n+2 + ... + a n+20 . Hence a n = a n+20 . Hence a 1 = a 281 = a 13 = a 273 = a 5 . So a n+4 = a n . We have to find a 210 = a 2 . So a 1 = a 17 = 3, a 3 = a 83 = 4, a 4 = a 144 = 9. We have 72 = a 1 + a 2 + ... + a 20 = 5a 1 + 5a 2 + 5a 3 + 5a 4 = 80 + 5a 2 . So a 2 = -8/5 42nd Swedish 2002 © John Scholes jscholes@kalva.demon.co.uk 21 December 2003 Last corrected/updated 21 Dec 03