2000 AIME II Problems/Problem 14
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Problem
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 
.
Contents |
Solution
Solution 1
Note that 
Thus for all 
,
So now,
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 
.
See also
| 2000 AIME II (Problems • Resources) | ||
| Preceded by Problem 13 | Followed by Problem 15 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
