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:
A.
B.
C.
D.
E.
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 , , or , with corresponding values , , or for , and , or 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