2002 Pan African MO Problems/Problem 1
Find all functions , (where is the set of all non-negative integers) such that for all and the minimum of the set is .
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 .
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