Problem:
The first 2007 positive integers are each written in base 3 . How many of these base-3 representations are palindromes? (A palindrome is a number that reads the same forward and backward.)
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Because , it is convenient to begin by counting the number of base-3 palindromes with at most 7 digits. There are two palindromes of length 1 , namely 1 and 2 . There are also two palindromes of length 2 , namely 11 and 22 . For , each palindrome of length is obtained by inserting one of the digits 0 , 1 , or 2 immediately after the th digit in a palindrome of length . Each palindrome of length is obtained by similarly inserting one of the strings 00,11 , or 22 . Therefore there are 6 palindromes of each of the lengths 3 and 4,18 of each of the lengths 5 and 6 , and 54 of length 7. Because the base-3 representation of 2007 is 2202100 , that integer is less than each of the palindromes 2210122, 2211122, 2212122, 2220222, 2221222 , and 2222222 . Thus the required total is .
The problems on this page are the property of the MAA's American Mathematics Competitions