2021 JMPSC Accuracy Problems/Problem 15
For all positive integers define the function to output For example, , , and Find the last three digits of
We can easily find that and so on. Thus, we claim that Now, we find we can easily find that
We proceed by induction. Our base case is Our inductive assumption is and we wish to prove that this pattern holds for
We can easily find that Using our inductive assumption, we obtain as desired.
~Solution by pinkpig, /wording fixes by samrocksnature
Solution 2 (More Algebraic)
We only care about the last digits, so we evaluate . Note the expression is simply , so factoring a we have . Now, we can divide by to get Evaluate the last digits to get ~Geometry285
The problems on this page are copyrighted by the Junior Mathematicians' Problem Solving Competition.