2012 USAMO Problems/Problem 4
Find all functions (where is the set of positive integers) such that for all positive integers and such that divides for all distinct positive integers , .
By the first condition we have and , so or and similarly for . By the second condition, we have for all positive integers .
Suppose that for some we have . We claim that for all . Indeed, from Equation (1) we have , and this is only possible if ; the claim follows by induction.
We now divide into cases:
This gives always from the previous claim, which is a solution.
This implies for all , but this does not satisfy the initial conditions. Indeed, we would have and so , a contradiction.
Case 3: ,
We claim always by induction. The base cases are and . Fix and suppose that . By Equation (1) we have that This implies (otherwise ). Also we have so . This gives the solutions and . The first case is obviously impossible, so , as desired. By induction, for all . This also satisfies the requirements.
We claim by a similar induction. Again if , then by (1) we have and so . Also note that and so . Then the only possible solution is . By induction, for all , and this satisfies all requirements.
In summary, there are three solutions: .
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