2000 AIME II Problems/Problem 14
Every positive integer has a unique factorial base expansion , meaning that , where each is an integer, , and . Given that is the factorial base expansion of , find the value of .
Thus for all ,
Therefore we have , if for some , and for all other .
Therefore we have:
Solution 2 (less formality)
Let . Note that since (or is significantly smaller than ), it follows that . Hence . Then , and as , it follows that . Hence , and we now need to find the factorial base expansion of
Since , we can repeat the above argument recursively to yield , and so forth down to . Now , so .
The remaining sum is now just . We can repeatedly apply the argument from the previous two paragraphs to find that , and if for some , and for all other .
Now for each , we have . Thus, our answer is .
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