28th CanMO 1996 Problem 5 Let x 1 , x 2 , ... , x m be positive rationals with sum 1. What is the maximum and minimum value of n - [n x 1 ] - [n x 2 ] - ... - [n x m ] for positive integers n?
[n x i ] ≤ n x i and n(x 1 + ... + x m ) = n, so clearly the value cannot be less than 0. But we can achieve 0 by taking n to be (for example) the product of the denominators of the x i . So the minimum value is 0. If we take n to be one less than the product of the denominators, then every term has its absolute value reduced by 1, so we get a total value of m-1. But each of (n x i - [n x i ]) is strictly less than one, so n - [n x 1 ] - ... - [n x m ] is less than m. It is an integer, so it is at most m - 1. Hence m - 1 is the maximum value. 28th CanMO 1996 © John Scholes jscholes@kalva.demon.co.uk 15 June 2002