2008 Indonesia MO Problems/Problem 8
Solution 1
Since , we know that .
Let , be , , respectively. Then, .
Let , be , , respectively. Then,
Let , be , , respectively. Then,
Let , be , , respectively. Then,
From the last 2 equations, we get that
Since , substituting, we get
If we take modulo of f(2) on both sides, we get
Because , we also know that . If , then .
Suppose :
since , we have . Or that . Thus, Thus, or .
case 1:
Let , and be an arbitrary integer . Then, Thus, .
case 2:
Let , and be an arbitrary integer . Then, This forms a linear line where Thus,
Upon verification for , we get
Upon verification for , we get
Thus, both equations, and are valid