24th IMO 1983 shortlist Problem 20 Find the greatest integer not exceeding 1 + 1/2 k + 1/3 k + ... + 1/N k , where k = 1982/1983 and N = 2 1983 .
We have (a 1983 - b 1983 ) = (a - b)(a 1982 + a 1981 b + a 1980 b 2 + ... + b 1982 ). So if a > b, we have (a 1983 - b 1983 )  (a - b) 1983 b 1982 , so (1/(m-1)) k > 1983 (m 1/1983 - (m-1) 1/1983 ). Summing gives 1 + 1/2 k + 1/3 k + ... + 1/(N-1) k > 1983(2 - 1) = 1983. Hence 1 + 1/2 k + 1/3 k + ... + 1/N k lies strictly between 1983 and 1984. So the integer part is 1983. 24th IMO shortlist 1983 © John Scholes jscholes@kalva.demon.co.uk 26 Nov 2003 Last corrected/updated 26 Nov 03