2004 AIME I Problems/Problem 15
Problem
For all positive integers , let
\[f(x)=\begin{cases}1 & \text{if x = 1}}\\ \frac x{10} & \text{if x is divisible by 10}\\ x+1 & \text{otherwise}\end{cases}\] (Error compiling LaTeX. Unknown error_msg)
and define a sequence as follows: and
for all positive integers
. Let
be the smallest
such that
. (For example,
and
.) Let
be the number of positive integers
such that
. Find the sum of the distinct prime factors of
.
Solution
We backcount the number of ways. Namely, we start at , which can only be reached if
, and then we perform
operations that either consist of
or
. We represent these operations in a string format, starting with the operation that sends
and so forth downwards. There are
ways to pick the first
operations; however, not all
of them may be
otherwise we return back to
, contradicting our assumption that
was the smallest value of
. Using complementary counting, we see that there are only
ways.
Since we performed the operation at least once in the first
operations, it follows that
, so that we no longer have to worry about reaching
again. Thus the remaining
operations can be picked in
ways, with a total of
strings.
However, we must also account for a sequence of or more
s in a row, because that implies that at least one of those numbers was divisible by
, yet the
was never used, contradiction. We must use complement counting again by determining the number of strings of
s of length
such that there are
s in a row. The first ten are not included since we already accounted for that scenario above, so our string of
s must be preceded by a
. There are no other restrictions on the remaining seven characters. Letting
to denote either of the functions, and
to indicate that the character appears
times in a row, then our bad strings can take the forms:
&\underbrace{BA^{[10]}}\square \square \square \square \square \square \square \square \\ &\square \underbrace{BA^{[10]}}\square \square \square \square \square \square \square \\ &\qquad \quad \cdots \quad \cdots \\ &\square \square \square \square \square \square \square \underbrace{BA^{[10]}} \square \\ &\square \square \square \square \square \square \square \square \underbrace{BA^{[10]}} \\
\end{align*}$ (Error compiling LaTeX. Unknown error_msg)There are ways to select the operations for the
s, and
places to place our
block. Thus, our answer is
, and the answer is
.
See also
2004 AIME I (Problems • Answer Key • Resources) | ||
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