1988 AIME Problems/Problem 8
The function , defined on the set of ordered pairs of positive integers, satisfies the following properties: Calculate .
Solution 1 (Algebra)
Let . By the substitution we rewrite the third property in terms of and then solve for Using the properties of we have ~MRENTHUSIASM (credit given to AoPS)
Solution 2 (Algebra)
Since all of the function's properties contain a recursive definition except for the first one, we know that in order to obtain an integer answer. So, we have to transform to this form by exploiting the other properties. The second one doesn't help us immediately, so we will use the third one.
Repeating the process several times,
Solution 3 (Number Theory)
Notice that satisfies all three properties:
For the first two properties, it is clear that and .
For the third property, using the identities and gives Hence, is a solution to the functional equation.
Since this is an AIME problem, there is exactly one correct answer, and thus, exactly one possible value of .
Therefore, we have
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