4th Australian 1983 Problem A1 Consider the following sequence: 1/1, 2/1, 1/2, 3/1, 2/2, 1/3, 4/1, 3/2, 2/3, 1/4, 5/1, ... , where we list all m/n with m+n = k in order of decreasing m, and then all m/n with m+n = k+1 etc. Each rational appears many times. Find the first five positions of 1/2. What is the position for the nth occurrence of 1/2? Find an expression for the first occurrence of p/q where p < q and p and q are coprime.
3, 14, 34, 63, 101 (9n 2 - 5n + 2)/2 (p+q-2)(p+q-1)/2 + q Solution There are k-1 terms with sum k. 1/2 appears as h/2h, so it appears once for each k which is a multiple of 3. For k = 3 it is the last term with k = 3, for k = 4 it is the penultimate term and so on. Thus the nth occurrence is in position (1+2) + (3+4+5) + (6+7+8) + ... + 9n-6 - (n-1) = n(9n-3)/2 - n + 1 = (9n 2 - 5n + 2)/2. p/q occurs in the row with k = p+q. The previous rows have 1 + 2 + ... + (p+q-2) = (p+q-2)(p+q-1)/2 terms. So it first occurs in position (p+q-2)(p+q-1)/2 + q. Solutions are also available in Australian Mathematical Olympiads 1979-1995 by H Lausch and P J Taylor, ISBN 0858896451, published by Australian Mathematics Trust, can be ordered by email. 4th Aus 1983 © John Scholes jscholes@kalva.demon.co.uk 2 January 2004 Last updated/corrected 2 Jan 04