2000 AIME II Problems/Problem 14
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
.
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 • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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