42nd Swedish 2002 Problem 4 For which integers n ≥8 is n 1/(n-7) an integer?
8, 9 Solution We claim that 2 n > n+7 for n ≥ 4. It is true for n = 4. Suppose it is true for n, then 2 n+1 = 2 n + 2 n > 1 + n + 7, so it is true for n+1 and hence for all n ≥ 4. Hence for n ≥ 11, n 1/(n-7)  1 for n ≥ 11, so n 1/(n-7) cannot be an integer for n ≥ 11. Checking we find it is for 8, 9 and is not for 10. 42nd Swedish 2002 © John Scholes jscholes@kalva.demon.co.uk 21 December 2003 Last corrected/updated 21 Dec 03