Let f(x) be a function defined on set of non negative integers and taking values in the same set such that f satisfies:
x – f(x) = 19[x/19] – 90[f(x)/90] and, 1900< f(1990) < 2000.
Determine the possible value(s) that f(1990) can take.
NOTE: [n] denotes the greatest integer < = n
Rearrange the terms to
90[f(x)/90]  f(x) = 19[x/19]  x
and substitute
90[f(1990)]  f(1990) = 1976  1990 = 14
What this says is that f(1990) is 14 more than a multiple of 90, or f(1990) = 90 k + 14, where k is an integer.
The lower limit of possible answers, 1900, is 10 more than a multiple of 90, so 1904 is 14 more than a multiple of 90, and so is the first possibility. The next possibility would be 1904+90 = 1994.
Thus the two possibilities are 1904 and 1994.

Posted by Charlie
on 20070428 10:10:39 