Problem:
Let f be a function satisfying f(xy)=f(x)/y for all positive real numbers x and y. If f(500)=3, what is the value of f(600) ?
Answer Choices:
A. 1
B. 2
C. 25
D. 3
E. 518
Solution:
Note that
f(600)=f(500⋅56)=6/5f(500)=6/53=25
OR
For all positive x,
f(x)=f(1⋅x)=xf(1)
so xf(x) is the constant f(1). Therefore,
600f(600)=500f(500)=500(3)=1500
so f(600)=6001500=25. Note. f(x)=x1500 is the unique function satisfying the given conditions.
The problems on this page are the property of the MAA's American Mathematics Competitions