Determine all possible nonzero real R satisfying R = [R]*{R}, such that 5*{R} - [R]/4 is an integer.

Note: [x] is the greatest integer ≤ x, and {x} = x - [x].

Denoting [R] by n and {R} by x, the constraints become:
            n + x = nx         (1)
            5x - n/4 = k       (2)        where k is an integer.

From (1)            x = n/(n - 1) and, since 0 <= x <1, n must be negative.   
Substituting in (2)
                        5n/(n - 1) - n/4 = k
Thus                 k = n(21 - n)/[4(n - 1)]

Since the consecutive integers n - 1 and n can have no common factors, it follows that n - 1 must be a factor of 21 - n, so that
(21 - n)/(n - 1) = m, for some negative integer m.
This can be written as n = 1 + 20/(m + 1), showing that m + 1 must be a factor of 20. The only possibilities are m + 1 = -1, -2, -4, -5, -10, -20.

Only two of these produce integer values for k:
m + 1 = -20 gives k = 0, n = 0 but leads to R = 0, which the question disallows.   
m + 1 = -4 gives k = 5, n = -4, x = 4/5 which leads to the only solution: R = -3.2

Posted by Harry on 2009-09-12 22:26:13