35th Swedish 1995 Problem 1 The pages of a booklet are numbered from 1 to 2n. A single sheet (of 2 pages) is removed. The numbers of the remaining pages sum to 963. How many pages did the booklet have originally and which pages were removed?
44 pages, pages 13/14 removed Solution The original total is n(2n+1). If the sheet with pages 2r-1, 2r is removed, then the total is 2n 2 +n-4r+1 = 963. So n must be even. If n ≤ 20, then 2n 2 +n ≤ 820  962 + 4r. Contradiction. So we must have n = 22. That gives r = 7, which works. 35th Swedish 1995 © John Scholes jscholes@kalva.demon.co.uk 16 January 2004 Last corrected/updated 16 Jan 04