8th Chinese 1993 Problem A2 α > 0 is real and n is a positive integer. What is the maximum possible value of α a + α b + α c + ... , where a, b, c, ... is any sequence of positive integers with sum n?
If α ≤ 1, then we want as many terms as possible and we want each exponent as small as possible. So we must take a = b = c = .... = 1 giving sum nα. If α > 1, then α(α a-1 - 1)(α b-1 ) > 1 for a, b > 1 and hence α a + α b  1. So a maximal expression has at most one exponent greater than 1, in other words it has the form mα + α n-m . Note that α n-1 (α - 1) is an increasing function of n. So for some N(α) we have α k-1 (α - 1) ≥ 1, and hence α + α k ≤ α k+1 iff k ≥ N(α). Hence the maximum value of mα + α n-m as m varies occurs at m = 0 or m = n. Thus the maximum value is the greater of nα and α n . If α ≥ 2, then 2α ≤ α 2 and nα  2, so the maximum is always α n . For 1 < α < 2, the maximum is nα if α < n 1/(n-1) and α n for α ≥ n 1/(n-1) . 8th Chinese 1993 © John Scholes jscholes@kalva.demon.co.uk 25 June 2002