2004 AMC 12B Problems/Problem 25
Problem
Given that is a
-digit number whose first digit is
, how many elements of the set
have a first digit of
?
Solution
Given digits, there must be exactly one power of
with
digits such that the first digit is
. Thus
contains
elements with a first digit of
. For each number in the form of
such that its first digit is
, then
must either have a first digit of
or
, and
must have a first digit of
. Thus there are also
numbers with first digit
and
numbers with first digit
. By using complementary counting, there are
elements of
with a first digit of
. Now,
has a first of
if and only if the first digit of
is
, so there are
elements of
with a first digit of
.
Alternate Solution
We can make the following chart for the possible loops of leading digits:
Thus each loop from can either have
or
numbers. Let there be
of the sequences of
numbers, and let there be
of the sequences of
numbers. We note that a
appears only in the loops of
, and also we are given that
has
digits.
Solving gives
and
, thus the answer is
.
--Az phx brandon jiang 19:49, 26 July 2014 (EDT)
See also
2004 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 24 |
Followed by Last problem |
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All AMC 12 Problems and Solutions |
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