Problem:
For a positive integer and nonzero digits , and , let be the -digit integer each of whose digits is equal to ; let be the -digit integer each of whose digits is equal to ; and let be the -digit (not -digit) integer each of whose digits is equal to . What is the greatest possible value of for which there are at least two values of such that
Answer Choices:
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Solution:
The equation is equivalent to
Dividing by and clearing fractions yields
As this must hold for two different values and , there are two such equations, and subtracting them gives
The second factor is non-zero, so and thus . From this it follows that and . Hence digit must be 3,6 , or 9 , with corresponding values 1,4 , or 9 for , and 2,8 , or 18 for . The case is invalid, so there are just two triples of possible values for , and , namely and . In fact, in these cases, for all positive integers ; for example, . The second triple has the greater coordinate sum, .
The problems on this page are the property of the MAA's American Mathematics Competitions