7th Australian 1986 Problem A2 What is the smallest tower of 100s that exceeds a tower of 100 threes? In other words, let a 1 = 3, a 2 = 3 3 , and a n+1 = 3 a n . Similarly, b 1 = 100, b 2 = 100 100 etc. What is the smallest n for which b n > a 100 ?
99 Solution We have 3 3  6b n-1 . Obviously 3 3 3 > 600 so it is true for n = 2. Suppose it is true for n. Note that 3 6 = 729 > 600, so a n+2 > 3 6b n-1 > 600 b n-1 = 6 b n-1 b n > 6b n . Solutions are also available in Australian Mathematical Olympiads 1979-1995 by H Lausch and P J Taylor, ISBN 0858896451, published by Australian Mathematics Trust. It can be ordered by email. 7th Aus 1986 © John Scholes jscholes@kalva.demon.co.uk 5 Jan 2004 Last corrected/updated 5 Jan 04