2004 AMC 12B Problems/Problem 25
Given that is a -digit number whose first digit is , how many elements of the set have a first digit of ?
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 .
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 .
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