2002 Pan African MO Problems/Problem 1
Problem
Find all functions , (where
is the set of all non-negative integers) such that
for all
and the minimum of the set
is
.
Solution
Let be the minimum of function
, so
. Substituting
means that
. Additionally, substituting
means that
, and substituting
means that
.
It seems as if for
. To prove this, we can use induction. The base case is covered because by substituting
, we get
. For the inductive step, assume that
. Substituting
means that
, so the inductive step holds.
Therefore, for
, and since
is increasing when
and
,
can not equal
if
. The only possible value of
left for
to equal
is
.
Since , the only function
that satisfies the requirements is
.
See Also
2002 Pan African MO (Problems) | ||
Preceded by First Problem |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All Pan African MO Problems and Solutions |