26th BMO 1990 Problem 2 The fractional part { x } of a real number is defined as x - [x]. Find a positive real x such that { x } + { 1/x } = 1 (*). Is there a rational x satisfying (*)?
There is no loss of generality in taking x > 1. So {1/x} = 1/x and {x} = x - n for some n. So we have x - n + 1/x = 1 or x 2 - (n+1)x + 1 = 0, so x = (n+1)/2 + 1/2 √(n 2 +2n-3). Evidently n = 0 and n = 1 do not work. But n = 2 gives x = (√5 + 3)/2, which works. Similarly, for example n = 3 gives x = 2 + √3, which works. But √(n 2 +2n-3) cannot be integral (because n is too small and n+1 is too large) and hence, by the usual argument, not rational either. So there are no rational solutions. 26th BMO 1990 © John Scholes jscholes@kalva.demon.co.uk 15 Aug 2002