2004 AMC 12B Problems/Problem 25
Contents
Problem
Given that is a -digit number whose first digit is , how many elements of the set have a first digit of ?
Solution 1
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 digit of if and only if the first digit of is , so there are elements of with a first digit of .
Solution 2
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 .
See also
2004 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 24 |
Followed by Last problem |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.