1997 AHSME Problems/Problem 29
Problem
Call a positive real number special if it has a decimal representation that consists entirely of digits and . For example, and are special numbers. What is the smallest such that can be written as a sum of special numbers?
Solution
Define a super-special number to be a number whose decimal expansion only consists of 's and 's. The problem is equivalent to finding the number of super-special numbers necessary to add up to . This can be done in numbers if we take Now assume for sake of contradiction that we can do this with strictly less than super-special numbers (in particular, less than .) Then the result of the addition won't have any carry over, so each digit is simply the number of super-special numbers which had a in that place. This means that in order to obtain the in , there must be super-special numbers, so the answer is .
See also
1997 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 28 |
Followed by Problem 30 | |
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