1996 IMO Problems/Problem 3
Problem
Let denote the set of nonnegative integers. Find all functions
from
to itself such that
Solution
Plugging in m = 0, we get f(f(n)) = f(n) .
With m = n = 0, we get f(0) = 0.
Let be the smallest fixed point of
such that
.
.
Plugging
, we get
.
By an easy induction, we get .
Let be another fixed point greater than
.
Let
, where
.
So, .
. But,
.
This means that the set of all fixed points of is
See Also
1996 IMO (Problems) • Resources | ||
Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 4 |
All IMO Problems and Solutions |