University of South Carolina High School Math Contest/1993 Exam/Problem 24
Problem
If and in general
then
![$\mathrm{(A) \ }3 \qquad \mathrm{(B) \ }1993 \qquad \mathrm{(C) \ }\frac 12 \qquad \mathrm{(D) \ }\frac 15 \qquad \mathrm{(E) \ } -2^{-1993}$](http://latex.artofproblemsolving.com/0/5/e/05e4773b2648f5283a450bb27e1fcda8639e663c.png)
Solution
. Then
,
and
.
It follows immediately that the function cycles and if
,
if
and
if
. Since
,
.
Alternatively, . Thus
, so
. Thus
so
.