2012 AMC 10B Problems/Problem 20
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 ?
Solution
Let's test each number starting from , for our first case , we start out with and the number is then given to Bernardo. He will double the given number so in this case . So now he gives this resulting number to Silvia and she adds to the number. So we have . Now this is then passed to Bernardo so he doubles this and we have . And now this is then given to Silvia so . So she passes it to Bernardo and we have . Silvia gets it and so we have . Continuing this we have . There is no need to continue with this because when Silvia adds