Problem:
Bernardo and Silvia play the following game. An integer between and , 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 to it and passes the result to Bernardo. The winner is the last person who produces a number less than . 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 . The smallest initial number for which he wins after two rounds is the smallest integer solution of , which is . Similarly, the smallest initial numbers for which he wins after three and four rounds are and , 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