IMO 1986 SL 2

Origin: SWE

Problem

Let f(x) = xn where n is a fixed positive integer and x = 1, 2, . . . . Is the decimal expansion a = 0.f(1)f(2)f(3) . . . rational for any value of n? The decimal expansion of a is defined as follows: If f(x) = d1(x)d2(x) . . . . . . dr(x)(x) is the decimal expansion of f(x), then a = 0.1d1(2)d2(2) . . . . . . dr(2)(2)d1(3) . . . dr(3)(3)d1(4) . . . .

Solution

No. If a were rational, its decimal expansion would be periodic from some point. Let p be the number of decimals in the period. Since f(102p) has 2np zeros, it contains a full periodic part; hence the period would consist only of zeros, which is impossible.