Problem:
Bernardo and Silvia play the following game. An integer between 0 and 999, inclusive, is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernardo. The winner is the last person who produces a number less than 1000 . Let be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of ?
Answer Choices:
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Solution:
The smallest initial number for which Bernardo wins after one round is the smallest integer solution of , which is 475 . The smallest initial number for which he wins after two rounds is the smallest integer solution of , which is 213 . Similarly, the smallest initial numbers for which he wins after three and four rounds are 82 and 16, respectively. There is no initial number for which Bernardo wins after more than four rounds. Thus , and the sum of the digits of is .
The problems on this page are the property of the MAA's American Mathematics Competitions