2012 USAMO Problems/Problem 4
Contents
Problem
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 , .
Solution
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:
Case 1:
This gives always from the previous claim, which is a solution.
Case 2:
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.
Case 4:
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: .
Summary
Three functions: since , and , fixed points on the function.
No elegant argument needed; so of course !
See Also
2012 USAMO (Problems • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
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